03-Gaussians.ipynb

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   "source": [
    "[Table of Contents](./table_of_contents.ipynb)"
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   "source": [
    "# Probabilities, Gaussians, and Bayes' Theorem"
   ]
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   "source": [
    "%matplotlib inline"
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   "source": [
    "#format the book\n",
    "import book_format\n",
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    "## Introduction\n",
    "\n",
    "The last chapter ended by discussing some of the drawbacks of the Discrete Bayesian filter. For many tracking and filtering problems our desire is to have a filter that is *unimodal* and *continuous*. That is, we want to model our system using floating point math (continuous) and to have only one belief represented (unimodal). For example, we want to say an aircraft is at (12.34, -95.54, 2389.5) where that is latitude, longitude, and altitude. We do not want our filter to tell us \"it might be at (1.65, -78.01, 2100.45) or it might be at (34.36, -98.23, 2543.79).\" That doesn't match our physical intuition of how the world works, and as we discussed, it can be prohibitively expensive to compute the multimodal case. And, of course, multiple position estimates makes navigating impossible.\n",
    "\n",
    "We desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is computationally efficient to calculate. Gaussian distributions provide all of these features."
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    "## Mean, Variance, and Standard Deviations\n",
    "\n",
    "Most of you will have had exposure to statistics, but allow me to cover this material anyway. I ask that you read the material even if you are sure you know it well. I ask for two reasons. First, I want to be sure that we are using terms in the same way. Second, I strive to form an intuitive understanding of statistics that will serve you well in later chapters. It's easy to go through a stats course and only remember the formulas and calculations, and perhaps be fuzzy on the implications of what you have learned.\n",
    "\n",
    "### Random Variables\n",
    "\n",
    "Each time you roll a die the *outcome* will be between 1 and 6. If we rolled a fair die a million times we'd expect to get a one 1/6 of the time. Thus we say the *probability*, or *odds* of the outcome 1 is 1/6. Likewise, if I asked you the chance of 1 being the result of the next roll you'd reply 1/6.  \n",
    "\n",
    "This combination of values and associated probabilities is called a [*random variable*](https://en.wikipedia.org/wiki/Random_variable). Here *random* does not mean the process is nondeterministic, only that we lack information about the outcome. The result of a die toss is deterministic, but we lack enough information to compute the result. We don't know what will happen, except probabilistically.\n",
    "\n",
    "While we are defining terms, the range of values is called the [*sample space*](https://en.wikipedia.org/wiki/Sample_space). For a die the sample space is {1, 2, 3, 4, 5, 6}. For a coin the sample space is {H, T}. *Space* is a mathematical term which means a set with structure. The sample space for the die is a subset of the natural numbers in the range of 1 to 6.\n",
    "\n",
    "Another example of a random variable is the heights of students in a university. Here the sample space is a range of values in the real numbers between two limits defined by biology.\n",
    "\n",
    "Random variables such as coin tosses and die rolls are *discrete random variables*. This means their sample space is represented by either a finite number of values or a countably infinite number of values such as the natural numbers. Heights of humans are called *continuous random variables* since they can take on any real value between two limits.\n",
    "\n",
    "Do not confuse the *measurement* of the random variable with the actual value. If we can only measure the height of a person to 0.1 meters we would only record values from 0.1, 0.2, 0.3...2.7, yielding 27 discrete choices. Nonetheless a person's height can vary between any arbitrary real value between those ranges, and so height is a continuous random variable. \n",
    "\n",
    "In statistics capital letters are used for random variables, usually from the latter half of the alphabet. So, we might say that $X$ is the random variable representing the die toss, or $Y$ are the heights of the students in the freshmen poetry class. Later chapters use linear algebra to solve these problems, and so there we will follow the convention of using  lower case for vectors, and upper case for matrices. Unfortunately these conventions clash, and you will have to determine which an author is using from context. I always use bold symbols for vectors and matrices, which helps distinguish between the two."
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    "## Probability Distribution\n",
    "\n",
    "\n",
    "The [*probability distribution*](https://en.wikipedia.org/wiki/Probability_distribution) gives the probability for the random variable to take any value in a sample space. For example, for a fair six sided die we might say:\n",
    "\n",
    "|Value|Probability|\n",
    "|-----|-----------|\n",
    "|1|1/6|\n",
    "|2|1/6|\n",
    "|3|1/6|\n",
    "|4|1/6|\n",
    "|5|1/6|\n",
    "|6|1/6|\n",
    "\n",
    "We denote this distribution with a lower case p: $p(x)$. Using ordinary function notation, we would write:\n",
    "\n",
    "$$P(X{=}4) = p(4) = \\frac{1}{6}$$\n",
    "\n",
    "This states that the probability of the die landing on 4 is $\\frac{1}{6}$. $P(X{=}x_k)$ is notation for \"the probability of $X$ being $x_k$\". Note the subtle notational difference. The capital $P$ denotes the probability of a single event, and the lower case $p$ is the probability distribution function. This can lead you astray if you are not observent. Some texts use $Pr$ instead of $P$ to ameliorate this. \n",
    "\n",
    "Another example is a fair coin. It has the sample space {H, T}. The coin is fair, so the probability for heads (H) is 50%, and the probability for tails (T) is 50%. We write this as\n",
    "\n",
    "$$\\begin{gathered}P(X{=}H) = 0.5\\\\P(X{=}T)=0.5\\end{gathered}$$\n",
    "\n",
    "Sample spaces are not unique. One sample space for a die is {1, 2, 3, 4, 5, 6}. Another valid sample space would be {even, odd}. Another might be {dots in all corners, not dots in all corners}. A sample space is valid so long as it covers all possibilities, and any single event is described by only one element.  {even, 1, 3, 4, 5} is not a valid sample space for a die since a value of 4 is matched both by 'even' and '4'.\n",
    "\n",
    "The probabilities for all values of a *discrete random value* is known as the *discrete probability distribution* and the probabilities for all values of a *continuous random value* is known as the *continuous probability distribution*.\n",
    "\n",
    "To be a probability distribution the probability of each value $x_i$ must be $x_i \\ge 0$, since no probability can be less than zero. Secondly, the sum of the probabilities for all values must equal one. This should be intuitively clear for a coin toss: if the odds of getting heads is 70%, then the odds of getting tails must be 30%. We formulize this requirement as\n",
    "\n",
    "$$\\sum\\limits_u P(X{=}u)= 1$$\n",
    "\n",
    "for discrete distributions, and as \n",
    "\n",
    "$$\\int\\limits_u P(X{=}u) \\,du= 1$$\n",
    "\n",
    "for continuous distributions.\n",
    "\n",
    "In the previous chapter we used probability distributions to estimate the position of a dog in a hallway. For example:"
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      "sum =  1.0\n"
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\n",
      "text/plain": [
       "<Figure size 800x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import kf_book.book_plots as book_plots\n",
    "\n",
    "belief = np.array([1, 4, 2, 0, 8, 2, 2, 35, 4, 3, 2])\n",
    "belief = belief / np.sum(belief)\n",
    "with book_plots.figsize(y=2):\n",
    "    book_plots.bar_plot(belief)\n",
    "print('sum = ', np.sum(belief))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each position has a probability between 0 and 1, and the sum of all equals one, so this makes it a probability distribution. Each probability is discrete, so we can more precisely call this a discrete probability distribution. In practice we leave out the terms discrete and continuous unless we have a particular reason to make that distinction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Mean, Median, and Mode of a Random Variable\n",
    "\n",
    "Given a set of data we often want to know a representative or average value for that set. There are many measures for this, and the concept is called a [*measure of central tendency*](https://en.wikipedia.org/wiki/Central_tendency). For example we might want to know the *average* height of the students in a class. We all know how to find the average of a set of data, but let me belabor the point so I can introduce more formal notation and terminology. Another word for average is the *mean*. We compute the mean by summing the values and dividing by the number of values. If the heights of the students in meters is \n",
    "\n",
    "$$X = \\{1.8, 2.0, 1.7, 1.9, 1.6\\}$$\n",
    "\n",
    "we compute the mean as\n",
    "\n",
    "$$\\mu = \\frac{1.8 + 2.0 + 1.7 + 1.9 + 1.6}{5} = 1.8$$\n",
    "\n",
    "It is traditional to use the symbol $\\mu$ (mu) to denote the mean.\n",
    "\n",
    "We can formalize this computation with the equation\n",
    "\n",
    "$$ \\mu = \\frac{1}{n}\\sum^n_{i=1} x_i$$\n",
    "\n",
    "NumPy provides `numpy.mean()` for computing the mean."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.8"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [1.8, 2.0, 1.7, 1.9, 1.6]\n",
    "np.mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a convenience NumPy arrays provide the method `mean()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.8"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array([1.8, 2.0, 1.7, 1.9, 1.6])\n",
    "x.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The *mode* of a set of numbers is the number that occurs most often. If only one number occurs most often we say it is a *unimodal* set, and if two or more numbers occur the most with equal frequency than the set is *multimodal*. For example the set {1, 2, 2, 2, 3, 4, 4, 4} has modes 2 and 4, which is multimodal, and the set {5, 7, 7, 13} has the mode 7, and so it is unimodal. We will not be computing the mode in this manner in this book, but we do use the concepts of unimodal and multimodal in a more general sense. For example, in the **Discrete Bayes** chapter we talked about our belief in the dog's position as a *multimodal distribution* because we assigned different probabilities to different positions.\n",
    "\n",
    "Finally, the *median* of a set of numbers is the middle point of the set so that half the values are below the median and half are above the median. Here, above and below is in relation to the set being sorted.  If the set contains an even number of values then the two middle numbers are averaged together.\n",
    "\n",
    "Numpy provides `numpy.median()` to compute the median. As you can see the median of {1.8, 2.0, 1.7, 1.9, 1.6} is 1.8, because 1.8 is the third element of this set after being sorted. In this case the median equals the mean, but that is not generally true."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.8"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.median(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expected Value of a Random Variable\n",
    "\n",
    "The [*expected value*](https://en.wikipedia.org/wiki/Expected_value) of a random variable is the average value it would have if we took an infinite number of samples of it and then averaged those samples together. Let's say we have $x=[1,3,5]$ and each value is equally probable. What value would we *expect* $x$ to have, on average?\n",
    "\n",
    "It would be the average of 1, 3, and 5, of course, which is 3. That should make sense; we would expect equal numbers of 1, 3, and 5 to occur, so $(1+3+5)/3=3$ is clearly the average of that infinite series of samples. In other words, here the expected value is the *mean* of the sample space.\n",
    "\n",
    "Now suppose that each value has a different probability of happening. Say 1 has an 80% chance of occurring, 3 has an 15% chance, and 5 has only a 5% chance. In this case we compute the expected value by multiplying each value of $x$ by the percent chance of it occurring, and summing the result. For this case we could compute\n",
    "\n",
    "$$\\mathbb E[X] = (1)(0.8) + (3)(0.15) + (5)(0.05) = 1.5$$\n",
    "\n",
    "Here I have introduced the notation $\\mathbb E[X]$ for the expected value of $x$. Some texts use $E(x)$. The value 1.5 for $x$ makes intuitive sense because $x$ is far more likely to be 1 than 3 or 5, and 3 is more likely than 5 as well.\n",
    "\n",
    "We can formalize this by letting $x_i$ be the $i^{th}$ value of $X$, and $p_i$ be the probability of its occurrence. This gives us\n",
    "\n",
    "$$\\mathbb E[X] = \\sum_{i=1}^n p_ix_i$$\n",
    "\n",
    "A trivial bit of algebra shows that if the probabilities are all equal, the expected value is the same as the mean:\n",
    "\n",
    "$$\\mathbb E[X] = \\sum_{i=1}^n p_ix_i = \\frac{1}{n}\\sum_{i=1}^n x_i = \\mu_x$$\n",
    "\n",
    "If $x$ is continuous we substitute the sum for an integral, like so\n",
    "\n",
    "$$\\mathbb E[X] = \\int_{a}^b\\, xf(x) \\,dx$$\n",
    "\n",
    "where $f(x)$ is the probability distribution function of $x$. We won't be using this equation yet, but we will be using it in the next chapter.\n",
    "\n",
    "We can write a bit of Python to simulate this. Here I take 1,000,000 samples and compute the expected value of the distribution we just computed analytically."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.50053"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "total = 0\n",
    "N = 1000000\n",
    "for r in np.random.rand(N):\n",
    "    if r <= .80: total += 1\n",
    "    elif r < .95: total += 3\n",
    "    else: total += 5\n",
    "\n",
    "total / N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can see that the computed value is close to the analytically derived value. It is not exact because getting an exact values requires an infinite sample size."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "What is the expected value of a die roll?\n",
    "\n",
    "### Solution\n",
    "\n",
    "Each side is equally likely, so each has a probability of 1/6. Hence\n",
    "$$\\begin{aligned}\n",
    "\\mathbb E[X] &= 1/6\\times1 + 1/6\\times 2 + 1/6\\times 3 + 1/6\\times 4 + 1/6\\times 5 + 1/6\\times6 \\\\\n",
    "&= 1/6(1+2+3+4+5+6)\\\\&= 3.5\\end{aligned}$$\n",
    "\n",
    "### Exercise\n",
    "\n",
    "Given the uniform continuous distribution\n",
    "\n",
    "$$f(x) = \\frac{1}{b - a}$$\n",
    "\n",
    "compute the expected value for $a=0$ and $b=20$.\n",
    "\n",
    "### Solution\n",
    "$$\\begin{aligned}\n",
    "\\mathbb E[X] &= \\int_0^{20}\\, x\\frac{1}{20} \\,dx \\\\\n",
    "&= \\bigg[\\frac{x^2}{40}\\bigg]_0^{20} \\\\\n",
    "&= 10 - 0 \\\\\n",
    "&= 10\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Variance of a Random Variable\n",
    "\n",
    "The computation above tells us the average height of the students, but it doesn't tell us everything we might want to know. For example, suppose we have three classes of students, which we label $X$, $Y$, and $Z$, with these heights:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = [1.8, 2.0, 1.7, 1.9, 1.6]\n",
    "Y = [2.2, 1.5, 2.3, 1.7, 1.3]\n",
    "Z = [1.8, 1.8, 1.8, 1.8, 1.8]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using NumPy we see that the mean height of each class is the same. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.8 1.8 1.8\n"
     ]
    }
   ],
   "source": [
    "print(np.mean(X), np.mean(Y), np.mean(Z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean of each class is 1.8 meters, but notice that there is a much greater amount of variation in the heights in the second class than in the first class, and that there is no variation at all in the third class.\n",
    "\n",
    "The mean tells us something about the data, but not the whole story. We want to be able to specify how much *variation* there is between the heights of the students. You can imagine a number of reasons for this. Perhaps a school district needs to order 5,000 desks, and they want to be sure they buy sizes that accommodate the range of heights of the students. \n",
    "\n",
    "Statistics has formalized this concept of measuring variation into the notion of [*standard deviation*](https://en.wikipedia.org/wiki/Standard_deviation) and [*variance*](https://en.wikipedia.org/wiki/Variance). The equation for computing the variance is\n",
    "\n",
    "$$\\mathit{VAR}(X) = \\mathbb  E[(X - \\mu)^2]$$\n",
    "\n",
    "Ignoring the square for a moment, you can see that the variance is the *expected value* for how much the sample space $X$ varies from the mean $\\mu:$ ($X-\\mu)$. I will explain the purpose of the squared term later. The formula for the expected value is $\\mathbb E[X] = \\sum\\limits_{i=1}^n p_ix_i$ so we can substitute that into the equation above to get\n",
    "\n",
    "$$\\mathit{VAR}(X) = \\frac{1}{n}\\sum_{i=1}^n (x_i - \\mu)^2$$\n",
    " \n",
    "Let's compute the variance of the three classes to see what values we get and to become familiar with this concept.\n",
    "\n",
    "The mean of $X$ is 1.8 ($\\mu_x = 1.8$) so we compute\n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "\\mathit{VAR}(X) &=\\frac{(1.8-1.8)^2 + (2-1.8)^2 + (1.7-1.8)^2 + (1.9-1.8)^2 + (1.6-1.8)^2} {5} \\\\\n",
    "&= \\frac{0 + 0.04 + 0.01 + 0.01 + 0.04}{5} \\\\\n",
    "\\mathit{VAR}(X)&= 0.02 \\, m^2\n",
    "\\end{aligned}$$\n",
    "\n",
    "NumPy provides the function `var()` to compute the variance:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.02 meters squared\n"
     ]
    }
   ],
   "source": [
    "print(f\"{np.var(X):.2f} meters squared\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is perhaps a bit hard to interpret. Heights are in meters, yet the variance is meters squared. Thus we have a more commonly used measure, the *standard deviation*, which is defined as the square root of the variance:\n",
    "\n",
    "$$\\sigma = \\sqrt{\\mathit{VAR}(X)}=\\sqrt{\\frac{1}{n}\\sum_{i=1}^n(x_i - \\mu)^2}$$\n",
    "\n",
    "It is typical to use $\\sigma$ for the *standard deviation* and $\\sigma^2$ for the *variance*. In most of this book I will be using $\\sigma^2$ instead of $\\mathit{VAR}(X)$ for the variance; they symbolize the same thing.\n",
    "\n",
    "For the first class we compute the standard deviation with\n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "\\sigma_x &=\\sqrt{\\frac{(1.8-1.8)^2 + (2-1.8)^2 + (1.7-1.8)^2 + (1.9-1.8)^2 + (1.6-1.8)^2} {5}} \\\\\n",
    "&= \\sqrt{\\frac{0 + 0.04 + 0.01 + 0.01 + 0.04}{5}} \\\\\n",
    "\\sigma_x&= 0.1414\n",
    "\\end{aligned}$$\n",
    "\n",
    "We can verify this computation with the NumPy method `numpy.std()` which computes the standard deviation. 'std' is a common abbreviation for standard deviation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "std 0.1414\n",
      "var 0.0200\n"
     ]
    }
   ],
   "source": [
    "print(f\"std {np.std(X):.4f}\")\n",
    "print(f\"var {np.std(X)**2:.4f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And, of course, $0.1414^2 = 0.02$, which agrees with our earlier computation of the variance.\n",
    "\n",
    "What does the standard deviation signify? It tells us how much the heights vary amongst themselves. \"How much\" is not a mathematical term. We will be able to define it much more precisely once we introduce the concept of a Gaussian in the next section. For now I'll say that for many things 68% of all values lie within one standard deviation of the mean. In other words we can conclude that for a random class 68% of the students will have heights between 1.66 (1.8-0.1414) meters and 1.94 (1.8+0.1414) meters. \n",
    "\n",
    "We can view this in a plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from kf_book.gaussian_internal import plot_height_std\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plot_height_std(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For only 5 students we obviously will not get exactly 68% within one standard deviation. We do see that 3 out of 5 students are within $\\pm1\\sigma$, or 60%, which is as close as you can get to 68% with only 5 samples. Let's look at the results for a class with 100 students.\n",
    "\n",
    ">  We write one standard deviation as  $1\\sigma$, which is pronounced \"one standard deviation\", not \"one sigma\". Two standard deviations is $2\\sigma$, and so on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean = 1.808\n",
      "std  = 0.133\n"
     ]
    }
   ],
   "source": [
    "from numpy.random import randn\n",
    "data = 1.8 + randn(100)*.1414\n",
    "mean, std = data.mean(), data.std()\n",
    "\n",
    "plot_height_std(data, lw=2)\n",
    "print(f'mean = {mean:.3f}')\n",
    "print(f'std  = {std:.3f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By eye roughly 68% of the heights lie within $\\pm1\\sigma$ of the mean 1.8, but we can verify this with code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "68.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum((data > mean-std) & (data < mean+std)) / len(data) * 100."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll discuss this in greater depth soon. For now let's compute the standard deviation for \n",
    "\n",
    "$$Y = [2.2, 1.5, 2.3, 1.7, 1.3]$$\n",
    "\n",
    "The mean of $Y$ is $\\mu=1.8$ m, so \n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "\\sigma_y &=\\sqrt{\\frac{(2.2-1.8)^2 + (1.5-1.8)^2 + (2.3-1.8)^2 + (1.7-1.8)^2 + (1.3-1.8)^2} {5}} \\\\\n",
    "&= \\sqrt{0.152} = 0.39 \\ m\n",
    "\\end{aligned}$$\n",
    "\n",
    "We will verify that with NumPy with"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "std of Y is 0.39 m\n"
     ]
    }
   ],
   "source": [
    "print(f'std of Y is {np.std(Y):.2f} m')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This corresponds with what we would expect. There is more variation in the heights for $Y$, and the standard deviation is larger.\n",
    "\n",
    "Finally, let's compute the standard deviation for $Z$. There is no variation in the values, so we would expect the standard deviation to be zero.\n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "\\sigma_z &=\\sqrt{\\frac{(1.8-1.8)^2 + (1.8-1.8)^2 + (1.8-1.8)^2 + (1.8-1.8)^2 + (1.8-1.8)^2} {5}} \\\\\n",
    "&= \\sqrt{\\frac{0+0+0+0+0}{5}} \\\\\n",
    "\\sigma_z&= 0.0 \\ m\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n"
     ]
    }
   ],
   "source": [
    "print(np.std(Z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we continue I need to point out that I'm ignoring that on average  men are taller than women. In general the height variance of a class that contains only men or women will be smaller than a class with both sexes. This is true for other factors as well. Well nourished children are taller than malnourished children. Scandinavians are taller than Italians. When designing experiments statisticians need to take these factors into account. \n",
    "\n",
    "I suggested we might be performing this analysis to order desks for a school district.  For each age group there are likely to be two different means - one clustered around the mean height of the females, and a second mean clustered around the mean heights of the males. The mean of the entire class will be somewhere between the two. If we bought desks for the mean of all students we are likely to end up with desks that fit neither the males or females in the school! \n",
    "\n",
    "We will not to consider these issues in this book.  Consult any standard probability text if you need to learn techniques to deal with these issues."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Why the Square of the Differences\n",
    "\n",
    "Why are we taking the *square* of the differences for the variance? I could go into a lot of math, but let's look at this in a simple way. Here is a chart of the values of $X$ plotted against the mean for $X=[3,-3,3,-3]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = [3, -3, 3, -3]\n",
    "mean = np.average(X)\n",
    "for i in range(len(X)):\n",
    "    plt.plot([i ,i], [mean, X[i]], color='k')\n",
    "plt.axhline(mean)\n",
    "plt.xlim(-1, len(X))\n",
    "plt.tick_params(axis='x', labelbottom=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we didn't take the square of the differences the signs would cancel everything out:\n",
    "\n",
    "$$\\frac{(3-0) + (-3-0) + (3-0) + (-3-0)}{4} = 0$$\n",
    "\n",
    "This is clearly incorrect, as there is more than 0 variance in the data. \n",
    "\n",
    "Maybe we can use the absolute value? We can see by inspection that the result is $12/4=3$ which is certainly correct — each value varies by 3 from the mean. But what if we have $Y=[6, -2, -3, 1]$? In this case we get $12/4=3$. $Y$ is clearly more spread out than $X$, but the computation yields the same variance. If we use the formula using squares we get a variance of 3.5 for $Y$, which reflects its larger variation.\n",
    "\n",
    "This is not a proof of correctness. Indeed, Carl Friedrich Gauss, the inventor of the technique, recognized that it is somewhat arbitrary. If there are outliers then squaring the difference gives disproportionate weight to that term. For example, let's see what happens if we have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variance of X with outlier    = 621.45\n",
      "Variance of X without outlier =   2.03\n"
     ]
    }
   ],
   "source": [
    "X = [1, -1, 1, -2, -1, 2, 1, 2, -1, 1, -1, 2, 1, -2, 100]\n",
    "print(f'Variance of X with outlier    = {np.var(X):6.2f}')\n",
    "print(f'Variance of X without outlier = {np.var(X[:-1]):6.2f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is this \"correct\"? You tell me. Without the outlier of 100 we get $\\sigma^2=2.03$, which accurately reflects how $X$ is varying absent the outlier. The one outlier swamps the variance computation. Do we want to swamp the computation so we know there is an outlier, or robustly incorporate the outlier and still provide an estimate close to the value absent the outlier? Again, you tell me. Obviously it depends on your problem.\n",
    "\n",
    "I will not continue down this path; if you are interested you might want to look at the work that James Berger has done on this problem, in a field called *Bayesian robustness*, or the excellent publications on *robust statistics* by Peter J. Huber [4]. In this book we will always use variance and standard deviation as defined by Gauss.\n",
    "\n",
    "The point to gather from this is that these *summary* statistics always tell an incomplete story about our data. In this example variance as defined by Gauss does not tell us we have a single large outlier. However, it is a powerful tool, as we can concisely describe a large data set with a few numbers. If we had 1 billion data points we would not want to inspect plots by eye or look at lists of numbers; summary statistics give us a way to describe the shape of the data in a useful way."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gaussians\n",
    "\n",
    "We are now ready to learn about [Gaussians](https://en.wikipedia.org/wiki/Gaussian_function). Let's remind ourselves of the motivation for this chapter.\n",
    "\n",
    "> We desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is computationally efficient to calculate.\n",
    "\n",
    "Let's look at a graph of a Gaussian distribution to get a sense of what we are talking about."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from filterpy.stats import plot_gaussian_pdf\n",
    "plot_gaussian_pdf(mean=1.8, variance=0.1414**2, \n",
    "                  xlabel='Student Height', ylabel='pdf');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This curve is a [*probability density function*](https://en.wikipedia.org/wiki/Probability_density_function) or *pdf* for short. It shows the relative likelihood  for the random variable to take on a value. We can tell from the chart student is somewhat more likely to have a height near 1.8 m than 1.7 m, and far more likely to have a height of 1.9 m vs 1.4 m. Put another way, many students will have a height near 1.8 m, and very few students will have a height of 1.4 m or 2.2 meters. Finally, notice that the curve is centered over the mean of 1.8 m.\n",
    "\n",
    "> I explain how to plot Gaussians, and much more, in the Notebook *Computing_and_Plotting_PDFs* in the \n",
    "Supporting_Notebooks folder. You can read it online [here](https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Supporting_Notebooks/Computing_and_plotting_PDFs.ipynb) [1].\n",
    "\n",
    "This may be recognizable to you as a 'bell curve'. This curve is ubiquitous because under real world conditions many observations are distributed in such a manner. I will not use the term 'bell curve' to refer to a Gaussian because many probability distributions have a similar bell curve shape. Non-mathematical sources might not be as precise, so be judicious in what you conclude when you see the term used without definition.\n",
    "\n",
    "This curve is not unique to heights — a vast amount of natural phenomena exhibits this sort of distribution, including the sensors that we use in filtering problems. As we will see, it also has all the attributes that we are looking for — it represents a unimodal belief or value as a probability, it is continuous, and it is computationally efficient. We will soon discover that it also has other desirable qualities which we may not realize we desire.\n",
    "\n",
    "To further motivate you, recall the shapes of the probability distributions in the *Discrete Bayes* chapter:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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UKWltbT3gYgEAoB4NrGbyzp07s2bNmsydO7fXeFtbW1avXv28626++eb893//d77+9a/n6quvftHzdHV1paurq+d427ZtSZLu7u50d3dXU/Irxt59Har72x96oAeJHtT7/vtbqX2u99dBve8/0YOkPnpQzd6qCu2bN2/O7t2709TU1Gu8qakpGzdu3Oeaxx57LHPnzs2qVasycOD+nW7hwoVZsGBBn/F77703Q4cOrabkV5yOjo5al1BzeqAHiR6Uu/831LqAl9WKFStqXcILKvd10D/qff+JHiSHdg927Nix33OrCu17NTQ09DquVCp9xpJk9+7dmTJlShYsWJDXv/71+/31r7jiirS3t/ccb9u2LaNGjUpbW1uGDRt2ICUXr7u7Ox0dHTn77LMzaNCgWpdTE3qgB4kelL7/+Xf8utYlvKzOOeecWpewT6W/Dg62et9/ogdJffRg790k+6Oq0H7MMcdkwIABfa6qb9q0qc/V9yTZvn17Hnzwwaxduzaf+MQnkiR79uxJpVLJwIEDc++99+bMM8/ss66xsTGNjY19xgcNGnTIftP2qoc9vhg90INED+p9//2l9B7X++ug3vef6EFyaPegmn1V9UbUwYMHp6Wlpc8/U3R0dGTChAl95g8bNiwPPfRQ1q1b1/Mxc+bMnHLKKVm3bl3e+c53VnN6AACoS1XfHtPe3p4LLrgg48ePT2tra5YuXZoNGzZk5syZSZ67teWpp57K1772tRx22GEZN25cr/XHHntshgwZ0mccAADYt6pD++TJk7Nly5ZcddVV6ezszLhx47JixYqMGTMmSdLZ2fmiz2wHAAD23wG9EXXWrFmZNWvWPj93yy23vODa+fPnZ/78+QdyWgAAqEtV/3IlAACgfwntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQuAMK7YsXL05zc3OGDBmSlpaWrFq16nnn3nXXXTn77LPzmte8JsOGDUtra2vuueeeAy4YAADqTdWhffny5ZkzZ07mzZuXtWvXZuLEiZk0aVI2bNiwz/n33Xdfzj777KxYsSJr1qzJu9/97px33nlZu3btSy4eAADqQdWh/dprr8306dMzY8aMjB07NosWLcqoUaOyZMmSfc5ftGhR/vZv/zZvf/vbc/LJJ+eaa67JySefnO9973svuXgAAKgHVYX2nTt3Zs2aNWlra+s13tbWltWrV+/X19izZ0+2b9+eo48+uppTAwBA3RpYzeTNmzdn9+7daWpq6jXe1NSUjRs37tfX+OIXv5hnn302559//vPO6erqSldXV8/xtm3bkiTd3d3p7u6upuRXjL37OlT3tz/0QA8SPaj3/fe3Uvtc76+Det9/ogdJffSgmr01VCqVyv5Ofvrpp3P88cdn9erVaW1t7Rn/3Oc+l1tvvTX/9V//9YLrb7/99syYMSPf/e53c9ZZZz3vvPnz52fBggV9xm+77bYMHTp0f8sF4GU2f9sbal3Cy2r+sEeqX6MHwMtkx44dmTJlSrZu3Zphw4a94NyqrrQfc8wxGTBgQJ+r6ps2bepz9f3/Wr58eaZPn55vfvObLxjYk+SKK65Ie3t7z/G2bdsyatSotLW1veiGXqm6u7vT0dGRs88+O4MGDap1OTWhB3qQ6EHp+59/x69rXcLL6pxzzql6jR4cfKX/OegPelAfPdh7N8n+qCq0Dx48OC0tLeno6Mhf/MVf9Ix3dHTk/e9///Ouu/322zNt2rTcfvvtOffcc1/0PI2NjWlsbOwzPmjQoEP2m7ZXPezxxeiBHiR6UO/77y96XHYP/DnQg+TQ7kE1+6oqtCdJe3t7LrjggowfPz6tra1ZunRpNmzYkJkzZyZ57ir5U089la997WtJngvsU6dOzZe+9KX86Z/+ac9V+sMPPzxHHXVUtacHAIC6U3Vonzx5crZs2ZKrrroqnZ2dGTduXFasWJExY8YkSTo7O3s9s/0rX/lKdu3aldmzZ2f27Nk94xdeeGFuueWWl74DAAA4xFUd2pNk1qxZmTVr1j4/93+D+I9+9KMDOQUAAPD/qfqXKwEAAP1LaAcAgMIJ7QAAUDihHQAACie0AwBA4YR2AAAonNAOAACFE9oBAKBwQjsAABTugH4jKkA9euutj/XTmd6Q+Xf8+qCfZe0FJx/0c3Do8ecAasOVdgAAKJzQDgAAhRPaAQCgcEI7AAAUTmgHAIDCCe0AAFA4oR0AAAontAMAQOGEdgAAKJzQDgAAhRPaAQCgcEI7AAAUTmgHAIDCCe0AAFC4gbUuAHjleOutj/XTmd6Q+Xf8+qCeYe0FJx/Urw8ALydX2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFE5oBwCAwgntAABQOKEdAAAKJ7QDAEDhhHYAACic0A4AAIUT2gEAoHBCOwAAFO6AQvvixYvT3NycIUOGpKWlJatWrXrB+StXrkxLS0uGDBmSE088MTfccMMBFQsAAPWo6tC+fPnyzJkzJ/PmzcvatWszceLETJo0KRs2bNjn/CeeeCLnnHNOJk6cmLVr1+bKK6/MJZdckjvvvPMlFw8AAPWg6tB+7bXXZvr06ZkxY0bGjh2bRYsWZdSoUVmyZMk+599www0ZPXp0Fi1alLFjx2bGjBmZNm1a/vEf//ElFw8AAPVgYDWTd+7cmTVr1mTu3Lm9xtva2rJ69ep9rvnJT36Stra2XmPvfe97s2zZsnR3d2fQoEF91nR1daWrq6vneOvWrUmS3/3ud+nu7q6m5FeM7u7u7NixI1u2bNlnT+qBHpTfg93/zzO1LuFls2XLlqrXHEr7T/Qg0YOk+h7U+/77S+n/P+gP9dCD7du3J0kqlcqLzq0qtG/evDm7d+9OU1NTr/GmpqZs3Lhxn2s2bty4z/m7du3K5s2bM2LEiD5rFi5cmAULFvQZb25urqZcgOd1zMdrXUHt6YEeJHpQ7/unDNu3b89RRx31gnOqCu17NTQ09DquVCp9xl5s/r7G97riiivS3t7ec7xnz5787ne/y/Dhw1/wPK9k27Zty6hRo/Kb3/wmw4YNq3U5NaEHepDoQb3vP9GDRA/qff+JHiT10YNKpZLt27dn5MiRLzq3qtB+zDHHZMCAAX2uqm/atKnP1fS9jjvuuH3OHzhwYIYPH77PNY2NjWlsbOw19upXv7qaUl+xhg0bdsi+MPeXHuhBogf1vv9EDxI9qPf9J3qQHPo9eLEr7HtV9UbUwYMHp6WlJR0dHb3GOzo6MmHChH2uaW1t7TP/3nvvzfjx4w/Z+5MAAODlVPXTY9rb23PjjTfmpptuyvr163PZZZdlw4YNmTlzZpLnbm2ZOnVqz/yZM2fmySefTHt7e9avX5+bbropy5Ytyyc/+cmXbxcAAHAIq/qe9smTJ2fLli256qqr0tnZmXHjxmXFihUZM2ZMkqSzs7PXM9ubm5uzYsWKXHbZZbn++uszcuTIfPnLX86HPvShl28Xh4DGxsZ85jOf6XNbUD3RAz1I9KDe95/oQaIH9b7/RA8SPfi/Gir784wZAACgZqq+PQYAAOhfQjsAABROaAcAgMIJ7QAAUDihvRCLFy9Oc3NzhgwZkpaWlqxatarWJfWb++67L+edd15GjhyZhoaGfOc736l1Sf1q4cKFefvb354jjzwyxx57bD7wgQ/k0UcfrXVZ/WrJkiV585vf3PMLNFpbW/P973+/1mXVzMKFC9PQ0JA5c+bUupR+NX/+/DQ0NPT6OO6442pdVr966qmn8tGPfjTDhw/P0KFD85a3vCVr1qypdVn95rWvfW2f10BDQ0Nmz55d69L6za5du/J3f/d3aW5uzuGHH54TTzwxV111Vfbs2VPr0vrN9u3bM2fOnIwZMyaHH354JkyYkAceeKDWZdWc0F6A5cuXZ86cOZk3b17Wrl2biRMnZtKkSb0enXkoe/bZZ3Paaafluuuuq3UpNbFy5crMnj07P/3pT9PR0ZFdu3alra0tzz77bK1L6zcnnHBC/v7v/z4PPvhgHnzwwZx55pl5//vfn4cffrjWpfW7Bx54IEuXLs2b3/zmWpdSE2984xvT2dnZ8/HQQw/VuqR+8/vf/z6nn356Bg0alO9///t55JFH8sUvfrFufiN48tzr/39///f+csYPf/jDNa6s/3z+85/PDTfckOuuuy7r16/PF77whfzDP/xD/umf/qnWpfWbGTNmpKOjI7feemseeuihtLW15ayzzspTTz1V69Jqq0LNveMd76jMnDmz19ipp55amTt3bo0qqp0klW9/+9u1LqOmNm3aVElSWblyZa1Lqak/+ZM/qdx44421LqNfbd++vXLyySdXOjo6Ku9617sql156aa1L6lef+cxnKqeddlqty6iZT33qU5Uzzjij1mUU5dJLL62cdNJJlT179tS6lH5z7rnnVqZNm9Zr7IMf/GDlox/9aI0q6l87duyoDBgwoPIv//IvvcZPO+20yrx582pUVRlcaa+xnTt3Zs2aNWlra+s13tbWltWrV9eoKmpp69atSZKjjz66xpXUxu7du3PHHXfk2WefTWtra63L6VezZ8/Oueeem7POOqvWpdTMY489lpEjR6a5uTkf+chH8vjjj9e6pH5z9913Z/z48fnwhz+cY489Nm9961vzz//8z7Uuq2Z27tyZr3/965k2bVoaGhpqXU6/OeOMM/Jv//Zv+eUvf5kk+cUvfpEf//jHOeecc2pcWf/YtWtXdu/enSFDhvQaP/zww/PjH/+4RlWVoerfiMrLa/Pmzdm9e3eampp6jTc1NWXjxo01qopaqVQqaW9vzxlnnJFx48bVupx+9dBDD6W1tTV//OMf86pXvSrf/va384Y3vKHWZfWbO+64I2vWrMmDDz5Y61Jq5p3vfGe+9rWv5fWvf31++9vf5uqrr86ECRPy8MMPZ/jw4bUu76B7/PHHs2TJkrS3t+fKK6/Mz372s1xyySVpbGzM1KlTa11ev/vOd76TP/zhD/nYxz5W61L61ac+9als3bo1p556agYMGJDdu3fnc5/7XP7qr/6q1qX1iyOPPDKtra357Gc/m7Fjx6apqSm33357/uM//iMnn3xyrcurKaG9EP/3KkKlUqmrKws85xOf+ET+8z//sy6vJpxyyilZt25d/vCHP+TOO+/MhRdemJUrV9ZFcP/Nb36TSy+9NPfee2+fq0v1ZNKkST3//aY3vSmtra056aST8tWvfjXt7e01rKx/7NmzJ+PHj88111yTJHnrW9+ahx9+OEuWLKnL0L5s2bJMmjQpI0eOrHUp/Wr58uX5+te/nttuuy1vfOMbs27dusyZMycjR47MhRdeWOvy+sWtt96aadOm5fjjj8+AAQPytre9LVOmTMnPf/7zWpdWU0J7jR1zzDEZMGBAn6vqmzZt6nP1nUPbxRdfnLvvvjv33XdfTjjhhFqX0+8GDx6c173udUmS8ePH54EHHsiXvvSlfOUrX6lxZQffmjVrsmnTprS0tPSM7d69O/fdd1+uu+66dHV1ZcCAATWssDaOOOKIvOlNb8pjjz1W61L6xYgRI/r8kDp27NjceeedNaqodp588sn867/+a+66665al9LvLr/88sydOzcf+chHkjz3A+yTTz6ZhQsX1k1oP+mkk7Jy5co8++yz2bZtW0aMGJHJkyenubm51qXVlHvaa2zw4MFpaWnpeYf8Xh0dHZkwYUKNqqI/VSqVfOITn8hdd92Vf//3f6/7v5T2qlQq6erqqnUZ/eI973lPHnrooaxbt67nY/z48fnrv/7rrFu3ri4De5J0dXVl/fr1GTFiRK1L6Renn356n8e9/vKXv8yYMWNqVFHt3HzzzTn22GNz7rnn1rqUfrdjx44cdljveDZgwIC6euTjXkcccURGjBiR3//+97nnnnvy/ve/v9Yl1ZQr7QVob2/PBRdckPHjx6e1tTVLly7Nhg0bMnPmzFqX1i+eeeaZ/OpXv+o5fuKJJ7Ju3bocffTRGT16dA0r6x+zZ8/Obbfdlu9+97s58sgje/7V5aijjsrhhx9e4+r6x5VXXplJkyZl1KhR2b59e+6444786Ec/yg9+8INal9YvjjzyyD7vYTjiiCMyfPjwunpvwyc/+cmcd955GT16dDZt2pSrr74627Ztq5uri5dddlkmTJiQa665Jueff35+9rOfZenSpVm6dGmtS+tXe/bsyc0335wLL7wwAwfWX0w577zz8rnPfS6jR4/OG9/4xqxduzbXXnttpk2bVuvS+s0999yTSqWSU045Jb/61a9y+eWX55RTTslFF11U69Jqq6bPrqHH9ddfXxkzZkxl8ODBlbe97W119bi/H/7wh5UkfT4uvPDCWpfWL/a19ySVm2++udal9Ztp06b1vP5f85rXVN7znvdU7r333lqXVVP1+MjHyZMnV0aMGFEZNGhQZeTIkZUPfvCDlYcffrjWZfWr733ve5Vx48ZVGhsbK6eeempl6dKltS6p391zzz2VJJVHH3201qXUxLZt2yqXXnppZfTo0ZUhQ4ZUTjzxxMq8efMqXV1dtS6t3yxfvrxy4oknVgYPHlw57rjjKrNnz6784Q9/qHVZNddQqVQqtflxAQAA2B/uaQcAgMIJ7QAAUDihHQAACie0AwBA4YR2AAAonNAOAACFE9oBAKBwQjsAABROaAcAgMIJ7QAAUDihHQAACie0AwBA4f5f9KYCA62XtoUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import kf_book.book_plots as book_plots\n",
    "belief = [0., 0., 0., 0.1, 0.15, 0.5, 0.2, .15, 0, 0]\n",
    "book_plots.bar_plot(belief)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They were not perfect Gaussian curves, but they were similar. We will be using Gaussians to replace the discrete probabilities used in that chapter!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nomenclature\n",
    "\n",
    "A bit of nomenclature before we continue - this chart depicts the *probability density* of a *random variable* having any value between ($-\\infty..\\infty)$. What does that mean? Imagine we take an infinite number of infinitely precise measurements of the speed of automobiles on a section of highway. We could then plot the results by showing the relative number of cars going past at any given speed. If the average was 120 kph, it might look like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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i3/+qA6mMf/sXbmhsNTpSjWKPfUIuTH3i0uXklH+o4UVNMH32I+htNtsFH0tfVvuylgO8/PLLLFy4kBUrVuDq6npJx506dSqTJk0q+T4jI4PQ0FCio6Px9vY+73byzywWC8uWLaN///5lfnoj9kd9opjNZmPiFztIyE7Cz93Mx+N60MjP/mb2sff+0GhHEhM+387vxx24tlcHBnc490KXvbH3PiHnUp+oPGdGt5RHhYr/gIAAHB0dz7nanpycfM5V+TOCg4PLbO/k5ES9evVKLZ85cyYvvPACv/76K+3bt7+k4wK4uLjg4uJyznKz2axOVkl0LuVs9t4n3lpxkB93JOHkYOKt26IIr2/fFxrstT/c2DmUvcnZvLXiII99s4uWDXxo21CzPIH99gk5P/WJS1eR81ehG36dnZ2Jioo65+OZZcuW0atXrzK36dmz5zntly5dSpcuXUoFnTFjBs899xw///wzXbp0ueTjiohUt9/3nODlX/YA8PTgNvSIqPcPW0hdNjm6JVe0DCS/0Mo9H24iJSv/nzcSEaliFZ7tZ9KkSbz33nvMnz+f2NhYJk6cSHx8POPGjQOKh9qMHj26pP24ceM4cuQIkyZNIjY2lvnz5zNv3jwmT55c0ubll1/miSeeYP78+TRp0oSkpCSSkpLIysoq93FFRIx0IDmLBxduxWaDkd3CuK17mNGRxGCODiZeG9GJiAAPjqfnMf6TzViKNP5fRIxV4TH/w4cPJzU1lWeffZbExETatm3LkiVLaNy4eAq7xMTEUnPvh4eHs2TJEiZOnMjs2bMJCQnh9ddfZ+jQoSVt5syZQ0FBAcOGDSt1rKeffppnnnmmXMcVETFKRp6Fez7cRGZ+IV2b+DFtcJsL3o8k9sPHzcw7o6O4cfYaNsSd4tnvd/PcjW2NjiUiduyibvgdP34848ePL3PdggULzlnWt29fNm/efN79HT58+JKPKyJiBJvNxpQvtnEoJZsQH1feui0KZ6eLeni61FHN6nsxa3hH7v5oEx+tO0K7hj7c0jXU6FgiYqf0F0pE5BK8u+oQv+w6gdnRxJzbogjwPHeSAZF+rYOY1K8FAE9+u5Ndx9MNTiQi9krFv4jIRVp/KJWXft4LwFPXt6FjqK+xgaRGu+/KZlzVqj75hVb+/fFm0nP1YCMRqX4q/kVELkJyRh73L9xCkdXGkE4NdYOv/CMHBxOv3tKBRn5uxJ/KYfIX20qeeyMiUl1U/IuIVJClyMr9n27hZGY+LYO8+O+QtrrBV8rF192ZOf/qjLOjA8t2n+DtPw4ZHUlE7IyKfxGRCprxy142HD6Fl4sTb93WGXfni5o7QexU+0a+PD24NQAv/7yHdYdSDU4kIvZExb+ISAX8vDORd/66Wjvj5g5EBHoanEhqo1u7hXFTp4ZYbXD/p1tIzsgzOpKI2AkV/yIi5XT0VA5TvtwOwD19IrimbbDBiaS2MplM/HdIO1oGeZGSlc/9C7dQqAeAiUg1UPEvIlIOliIr/1m4hcy8QjqH+TJlQEujI0kt5+bsyFu3dcbTxYkNcad4/fcDRkcSETug4l9EpBxm/rKXrUdP4+3qxOsjO2F21MunXLqIQE/+O6T4ib9v/L6fNQdTDE4kInWd/nqJiPyD5XuTS2ZlmXFzBxr5uRucSOqSGzo25JYujbDZYOKiraRm5RsdSUTqMBX/IiIXkJSex0OfbwPgjl5NGNBG4/yl8j0zuA3N6ntyIiOfyV9sw2rV/P8iUjVU/IuInEeR1caERVs4lV1AmxBvpg5qZXQkqaPcnZ1489ZOODs5sHzvSeb/GWd0JBGpo1T8i4icxxu/72fdoVN4ODvy5q2dcXFyNDqS1GGtgr156rri+f9f+nkP246eNjaQiNRJKv5FRMqw9mAqr/+2H4AXbmpHeICHwYnEHvyrexiD2gVjKbLxn4VbyMizGB1JROoYFf8iImdJyy5gwqItWG1wS5dG3NCxodGRxE6YTCam39Sehr5uxJ/K4bGvdmCzafy/iFQeFf8iIn9js9mY+tUOTmTkExHowTOD2xgdSeyMj5uZN27thJODiR+2J/JFTILRkUSkDlHxLyLyN19sSuDnXUmYHU28PqIT7s5ORkcSO9Q5zI9J0S0AmPbdLo6kZhucSETqChX/IiJ/iUvJ5pnvdwHwUHRL2jb0MTiR2LN7+zSlW7g/2QVFTFi0lcIiq9GRRKQOUPEvIgJYiqxM+GwLOQVF9Iyoxz29I4yOJHbO0cHE/4Z3xMvViS3xp3nj9wNGRxKROkDFv4gI8Nqv+9mWkI6Pm5lXbumAg4PJ6EgiNPR14/kb2wLFU8/GHEkzOJGI1HYq/kXE7m2IO8WcFcVXVV8Y0o4QXzeDE4n8vxs6NuTGjiFYbTBx0Vay8guNjiQitZiKfxGxa+m5FiYu2orVBsOiGnFt+wZGRxI5x7M3ti2Z/vOZ73YZHUdEajEV/yJi1576difHTucS5u+uaT2lxvJ2NfO/4R1xMMGXMQks2ZFodCQRqaVU/IuI3fpmyzG+3Xq85MZKTxdN6yk1V7dwf8Zf0QyAqV/tIDE91+BEIlIbqfgXEbt0/HQuT36zE4AHrmpOVGM/gxOJ/LMH+zWnQyMf0nMtPPT5NqxWPf1XRCpGxb+I2B2r1caUL7eRmV9Ix1Bf7ruyqdGRRMrF7OjA/4Z3xM3syJqDqby/5rDRkUSkllHxLyJ256N1R/jzQCquZgdevaUDTo56KZTaIyLQkyeuiwTg5Z/3cCA5y+BEIlKb6C+eiNiVQyezmP5TLABTB0YSEehpcCKRiru1Wxh9WwSSX2jloc/19F8RKT8V/yJiNwqLrEz6fBt5FiuXNwtgVI/GRkcSuSgmk4mXhrbH29WJbQnpzFlx0OhIIlJLqPgXEbsxd+VBth49jZerEy8Pa6+n+EqtFuzjynN/Pf339d/2s/NYusGJRKQ2UPEvInZh57F0Zv26H4Bpg9voKb5SJwzuEMKgdsEUWm1M+nwreZYioyOJSA2n4l9E6rz8wiIe+nwbhVYbA9oEMaRTQ6MjiVQKk8nE8ze2I8DThX0nsvjfsn1GRxKRGk7Fv4jUea8u28feE5kEeDrzwpB2mEwa7iN1h7+HMy/e1A6Ad1YdYuPhUwYnEpGaTMW/iNRpGw+f4p0/DgHwwpB21PN0MTiRSOXr1zqIm6MaYbPBQ59vIzu/0OhIIlJDqfgXkTorO7+Qhz7fhs0Gw6IaEd0m2OhIIlXmqetb09DXjfhTObywJNboOCJSQ6n4F5E6679LYok/lUNDXzeeur610XFEqpSXq5kZw9oD8Mn6eFbuO2lwIhGpiVT8i0idtHLfST5dHw/AjGHt8XY1G5xIpOr1ahbAHb2aAPDwl9tIz7EYG0hEahwV/yJS52TkWXh08XYA7ujVhF7NAgxOJFJ9HrmmFREBHpzIyOeZ73cZHUdEahgV/yJS50xfEktieh5h/u48fE1Lo+OIVCs3Z0deuaUDDib4essxlu0+YXQkEalBVPyLSJ3yx76TLNxwFICXh7XH3dnJ4EQi1a9TmB9394kA4LGvd3A6p8DgRCJSU6j4F5E6I/Os4T49IuoZnEjEOBP7taBpoAcnM/N59vvdRscRkRpCxb+I1BkvLNnDcQ33EQHA1ezIjJuLh/98teUYv2r4j4ig4l9E6ohV+0+ycEPx7D4a7iNSrHOYH3f3Lh7+M1XDf0QEFf8iUgcUD/fZAcDtPRtruI/I30zsr+E/IvL/VPyLSK03/ac9HDudS6i/Gw9f08roOCI1iob/iMjfqfgXkVpt9f6Ukod5vTy0Ax4uGu4jcra/D/957OsdeviXiB1T8S8itVZWfiGP/DW7z+iejenZVMN9RM5nYv8WRAR6kJyZz7Qf9PAvEXul4l9Eaq3pS2JLhvs8ouE+IhfkanZk5pnhP5s1/EfEXqn4F5Fa6c8DKXzy13Cfl4a213AfkXLoHObHWA3/EbFrF1X8z5kzh/DwcFxdXYmKimLVqlUXbL9y5UqioqJwdXUlIiKCuXPnllq/a9cuhg4dSpMmTTCZTMyaNeucfTzzzDOYTKZSX8HBwRcTX0Rquaz8Qh7+sni4z6gejenVNMDgRCK1xyQN/xGxaxUu/hctWsSECRN4/PHH2bJlC71792bgwIHEx8eX2T4uLo5BgwbRu3dvtmzZwmOPPcYDDzzA4sWLS9rk5OQQERHBiy++eMGCvk2bNiQmJpZ87dixo6LxRaQOePGn4uE+jfzceHSghvuIVISr2ZEZw/5/+M9vsRr+I2JPKlz8v/rqq9x1112MHTuWyMhIZs2aRWhoKG+99VaZ7efOnUtYWBizZs0iMjKSsWPHMmbMGGbOnFnSpmvXrsyYMYMRI0bg4uJy3mM7OTkRHBxc8hUYGFjR+CJSy605kMLH687M7qPhPiIXI6rx/w//mfqVhv+I2JMKFf8FBQXExMQQHR1danl0dDRr1qwpc5u1a9ee037AgAFs2rQJi6ViLzb79+8nJCSE8PBwRowYwaFDhyq0vYjUbln5hUz5a7jPbT3C6NVMw31ELtbfh/88+4Me/iViLyp0ySwlJYWioiKCgoJKLQ8KCiIpKanMbZKSkspsX1hYSEpKCg0aNCjXsbt3786HH35IixYtOHHiBM8//zy9evVi165d1KtX9vR++fn55Ofnl3yfkZEBgMViqfAbDyntzPnTeZQzqqNPvPDj7uLhPr6uTO7XTP2vBtNrRM3nCLx4YxuGv7eBxZsTGNA6kCtbVt0n6uoTcjb1icpTkXN4UZ+Xm0ymUt/bbLZzlv1T+7KWX8jAgQNL/r9du3b07NmTpk2b8sEHHzBp0qQyt5k+fTrTpk07Z/nSpUtxd3cv97Hl/JYtW2Z0BKlhqqpP7Es38eluRwBuCMlm5W9Lq+Q4Urn0GlHzXRHswPJEByYv2szUjkW4V/FIOvUJOZv6xKXLyckpd9sK/RMPCAjA0dHxnKv8ycnJ51zdPyM4OLjM9k5OTue9Yl8eHh4etGvXjv3795+3zdSpU0u9McjIyCA0NJTo6Gi8vb0v+thS/A5z2bJl9O/fH7PZbHQcqQGqsk9k5xcy4801QB4juzZiwuDWlbp/qXx6jag9rrIUMXj2WuJSc9hYGMZLg9tWyXHUJ+Rs6hOV58zolvKoUPHv7OxMVFQUy5YtY8iQISXLly1bxg033FDmNj179uT7778vtWzp0qV06dLlkn7R+fn5xMbG0rt37/O2cXFxKfMGYrPZrE5WSXQu5WxV0Sde+XEvCafzaOjrxuPXtcFs1k2+tYVeI2o+s9nMzFs6MGzuWr7acpzrOzTkylb1q/R46hPyd+oTl64i56/Cs/1MmjSJ9957j/nz5xMbG8vEiROJj49n3LhxQPHV9tGjR5e0HzduHEeOHGHSpEnExsYyf/585s2bx+TJk0vaFBQUsHXrVrZu3UpBQQHHjh1j69atHDhwoKTN5MmTWblyJXFxcaxfv55hw4aRkZHB7bffXtEfQURqkbUHU/lo3REAXh7WHk/N7iNS6aIa+3PXZeFA8ew/GXkagy1SV1X4r+jw4cNJTU3l2WefJTExkbZt27JkyRIaN24MQGJiYqk5/8PDw1myZAkTJ05k9uzZhISE8PrrrzN06NCSNsePH6dTp04l38+cOZOZM2fSt29fVqxYAUBCQgIjR44kJSWFwMBAevTowbp160qOKyJ1T05BIY8sLp7d59buYVym2X1EqsxD0S35bU8ycSnZ/PeHWF4a1t7oSCJSBS7qEtr48eMZP358mesWLFhwzrK+ffuyefPm8+6vSZMmJTcBn89nn31WoYwiUvu9/PNe4k/lEOLjylQ9zEukSrk5O/LysPbc8vZaFm06yqD2DejbQs/TEalrKjzsR0SkOmyIO8WCNYcBeHFoe7xcNR5UpKp1beLPHb2aAPDo4u0a/iNSB6n4F5EaJ7egiIe/3AbA8C6h9NHVR5FqM2VASxrXcycxPY/pS2KNjiMilUzFv4jUOK8s3cvh1ByCvV15/LpIo+OI2BV3ZydeHlo83n/hhqP8se+kwYlEpDKp+BeRGiXmyCnm/RkHwPSb2uGt4T4i1a57RL2S4T9Tv9pBpob/iNQZKv5FpMbIsxQx5cvt2GwwtHOjKp1rXEQu7OFrWhLm786x07lM/2mP0XFEpJKo+BeRGuN/v+7j0Mls6nu58NR1eoqviJHcnZ146a/hP5+uj2f1/hSDE4lIZVDxLyI1wtajp3n3j0MA/HdIO3zcNdxHxGg9m9ZjdM/i5+k8sng7WfmFBicSkUul4l9EDJdfWMSUL7ZhtcENHUPo3zrI6Egi8pdHrmlFIz83jp3O5cWfNPuPSG2n4l9EDPf6b/vZn5xFgKczz1zfxug4IvI3Hi7/P/vPx+viWXNAw39EajMV/yJiqB0J6cxdWTzc5/kb2+Ln4WxwIhE5W69mAfyrexgADy/eTraG/4jUWir+RcQwBYVWpny5jSKrjWvbN+Catg2MjiQi5zF1UCQNfd1ISMvl5Z81+49IbaXiX0QMM3v5AfYkZeLv4cyzgzXcR6Qm83T5/9l/Plh7hHWHUg1OJCIXQ8W/iBhi1/F0Zi8/AMC0wW2o5+licCIR+SeXNw9gZLe/hv98uZ2cAg3/EaltVPyLSLWzFFmZ8sV2Cq02rmkTzHXtNdxHpLZ4bFArQnxciT+Vw8s/7zU6johUkIp/Eal2c1ccZHdiBr7uZp67sS0mk8noSCJSTl6uZl78a/jPgjWHWa/hPyK1iop/EalWe5Myef33/QA8c30bAr003EektunTIpARXUOB4tl/cguKDE4kIuWl4l9Eqk1hkZXJX2zDUmSjX2R9bugYYnQkEblIj10bSQMfV46k5jDjFw3/EaktVPyLSLV5Z9UhdhxLx9vVif8OaafhPiK1mLermek3tQPg/TVxbDx8yuBEIlIeKv5FpFocSM5k1rLi4T5PXd+GIG9XgxOJyKW6omV9bo5qhM1WPPuPhv+I1Hwq/kWkyhVZbUz+YjsFRVauaBnI0M4NjY4kIpXkietaE+TtQlxKNq8s1fAfkZpOxb+IVLl5qw+x9ehpvFycmH6ThvuI1CU+bv8//Gfen3HEHNHwH5GaTMW/iFSpgyezeGXpPgAevzaSBj5uBicSkcp2VasghnYuHv4z5Yvt5Fk0/EekplLxLyJVpni4zzbyC630bh7A8L+mBhSRuuep61pT38uFQynZ/G/ZPqPjiMh5qPgXkSrz7qpDbIkvHu7z0tD2Gu4jUof5uJt5YUjx8J93Vx1ic3yawYlEpCwq/kWkSuw7kcmrfw33efL61oT4ariPSF3Xr3UQN3VqiNUGU77YpuE/IjWQin8RqXSWIisPfb6NgiIrV7UqngpQROzDU9e3JtDLhYMns5n1636j44jIWVT8i0ile2vFQXYcSy+ZBUTDfUTsh6+7c8nwn3f+OMjWo6eNDSQipaj4F5FKtTsxg9d/K77aN22wHuYlYo/6tw7ixo4hGv4jUgOp+BeRSlNohUcW76TQamNAmyBu6BhidCQRMcjT17chwNOF/clZJRcERMR4Kv5FpNL8kuDAnhNZ+Hs4898hGu4jYs/8PJx5/sa2AMxdeZDtCekGJxIRUPEvIpVke0I6vx4rLvafu6EtAZ4uBicSEaNd0zaY6zsUD/959OudFFqNTiQiKv5F5JLlWYp45KudWDFxbdtgrm3fwOhIIlJDTBvchgBPZ/YnZ7PkqMoOEaPpX6GIXLL//bqPAyez8TLbePr6VkbHEZEaxN/DmedvLJ795/fjJmKO6OFfIkZS8S8ilyTmSBrv/nEIgOERVvzcnQ1OJCI1zTVtgxnSKQQbJh7+aifZ+YVGRxKxWyr+ReSi5RYUMfmLbVhtMKRjA9r524yOJCI11JODWuLrbCP+VC4vLIk1Oo6I3VLxLyIXbcYve4lLySbI24XHB2m4j4icn5ermX81K77j95P18Szfm2xwIhH7pOJfRC7KmgMpzP8zDoCXhrbHx81scCIRqela+NgY3SMMgEe+3M7pnAKDE4nYHxX/IlJh6bkWJn+xDYCR3cK4omV9gxOJSG0xuX9zIgI9SM7M56lvdxkdR8TuqPgXkQqb9t0ujqfn0bieO09cG2l0HBGpRdycHXn1lo44Opj4bttxvt923OhIInZFxb+IVMiP2xP5assxHEzw6i0d8XBxMjqSiNQyHUN9ue+KpgA8+e1OTmTkGZxIxH6o+BeRckvOyOPxb3YAMP6KZkQ19jM4kYjUVvdf1Zy2Db05nWPhkcXbsdk0W5hIdVDxLyLlYrPZmPLldk7nWGjb0JsHrm5udCQRqcWcnRx49ZaOODs5sGLvST7beNToSCJ2QcW/iJTLx+vjWbnvJC5ODvzvrz/YIiKXokWQFw8PaAnAcz/sJj41x+BEInWf/nqLyD86dDKL//64G4BHB7aieZCXwYlEpK4Yc1k43cL9ySko4qEvtlJk1fAfkaqk4l9ELqiwyMrEz7eRZ7FyWbN63N6zidGRRKQOcXAw8crNHfBwdmTj4TTe/uOg0ZFE6jQV/yJyQbOXH2Tb0dN4uzox8+YOODiYjI4kInVMqL87Tw9uA8CrS/exIyHd4EQidZeKfxE5r21HT/P67/sBeO7GtjTwcTM4kYjUVTdHNWJg22AKrTYe/GwLOQWFRkcSqZNU/ItImXILipi4qHj87XXtG3BDx4ZGRxKROsxkMjH9pnYEe7tyKCWb53+MNTqSSJ2k4l9EyvT8j7s5lJJNkLcLz9/Y1ug4ImIHfN2defWWDphM8On6eJbuSjI6kkido+JfRM7xy64kPlkfD8ArN3fE193Z4EQiYi96NQvg7t4RADyyeDvJevqvSKW6qOJ/zpw5hIeH4+rqSlRUFKtWrbpg+5UrVxIVFYWrqysRERHMnTu31Ppdu3YxdOhQmjRpgslkYtasWZVyXBGpuKT0PB5ZvB2Ae/tEcHnzAIMTiYi9eSi6Ba0beJOWY+GhL7Zh1fSfIpWmwsX/okWLmDBhAo8//jhbtmyhd+/eDBw4kPj4+DLbx8XFMWjQIHr37s2WLVt47LHHeOCBB1i8eHFJm5ycHCIiInjxxRcJDg6ulOOKSMVZrTYmfb615Cm+D0W3NDqSiNghFydHXh/ZERcnB1btT2HBmsNGRxKpMypc/L/66qvcddddjB07lsjISGbNmkVoaChvvfVWme3nzp1LWFgYs2bNIjIykrFjxzJmzBhmzpxZ0qZr167MmDGDESNG4OLiUinHFZGKe2fVIdYcTMXN7MjrIzrpKb4iYphm9b144tpIAF78eQ97kjIMTiRSNzhVpHFBQQExMTE8+uijpZZHR0ezZs2aMrdZu3Yt0dHRpZYNGDCAefPmYbFYMJvNVXJcgPz8fPLz80u+z8gofuGwWCxYLJZ/PK6c35nzp/NYd+w4ls7MX/YC8OS1rQj1danQ71d9Qv5O/UHOdjF9YnhUCL/FnmDFvhQe+HQLX43rjovZsaoiSjXT60Tlqcg5rFDxn5KSQlFREUFBQaWWBwUFkZRU9h35SUlJZbYvLCwkJSWFBg0aVMlxAaZPn860adPOWb506VLc3d3/8bjyz5YtW2Z0BKkE+UXw8nZHCq0mOtaz4p60jSVLtl3UvtQn5O/UH+RsFe0T/bxhk9mRfclZ3PfOMm4Kt1ZRMjGKXicuXU5OTrnbVqj4P8NkKv2ET5vNds6yf2pf1vLKPu7UqVOZNGlSyfcZGRmEhoYSHR2Nt7d3hY4tpVksFpYtW0b//v3L9emN1GyPfr2TlLzjNPBx5b17e+LjVvHfqfqE/J36g5ztUvpEcORJxn60hZVJDvyrXxRXtgysopRSnfQ6UXnOjG4pjwoV/wEBATg6Op5ztT05Ofmcq/JnBAcHl9neycmJevXqVdlxAVxcXMq8h8BsNquTVRKdy9rv+23HWbz5OA4meG1EJwK8L+1TMfUJ+Tv1BznbxfSJfm1CuPOyNN7/8zCPfLWTnx7sQ7CPaxUllOqm14lLV5HzV6G7+ZydnYmKijrn45lly5bRq1evMrfp2bPnOe2XLl1Kly5dyh30Yo4rIv8sIS2Hx77eAcD9VzajW7i/wYlERMr26MBWtG1YPP3ng59toUjTf4pclApP5TFp0iTee+895s+fT2xsLBMnTiQ+Pp5x48YBxUNtRo8eXdJ+3LhxHDlyhEmTJhEbG8v8+fOZN28ekydPLmlTUFDA1q1b2bp1KwUFBRw7doytW7dy4MCBch9XRCqmsMjKhM+2kplXSOcwXx64urnRkUREzsvFyZE3RnbGw9mR9XGneOP3/UZHEqmVKjzmf/jw4aSmpvLss8+SmJhI27ZtWbJkCY0bNwYgMTGx1Nz74eHhLFmyhIkTJzJ79mxCQkJ4/fXXGTp0aEmb48eP06lTp5LvZ86cycyZM+nbty8rVqwo13FFpGL+9+s+Nh1Jw9PFiddGdMLJUdN6ikjNFh7gwX+HtGPCoq28/tt+ekTUo0dE+YYQi0ixi7rhd/z48YwfP77MdQsWLDhnWd++fdm8efN599ekSZOSm4Av9rgiUn5/7DvJnBUHAXhxaDtC/TX7lYjUDjd2asjqAyl8GZPAhM+2suTB3vh7OBsdS6TW0KU+ETtzIiOPiYu2YrPBv7qHcV37EKMjiYhUyLTBbYgI9CApI48pX2wr1wVEESmm4l/EjhQWWXlg4RZSswuIbODNk9e1NjqSiEiFebg48ebIzjg7OfDbnmTm/3nY6EgitYaKfxE78vpv+1kfdwoPZ0dm39oJVz0pU0RqqdYh3jx5bSQAL/4Uy46EdIMTidQOKv5F7MTq/Sm8sbx4Bq0XbmpHRKCnwYlERC7NbT0aM6BNEJYiG/cv3ExGnsXoSCI1nop/ETuQnJnHhL/G+Y/sFsoNHRsaHUlE5JKZTCZeHtqBhr5uHEnN0fh/kXJQ8S9SxxVZbUz4bCspWfm0Cvbi6evbGB1JRKTS+LibmfOvzpgdTfyy6wTzVscZHUmkRlPxL1LHvfn7AdYcTMXd2ZE3b+2scf4iUud0CPXlqb8mMJj+0x42Hj5lcCKRmkvFv0gdtmr/SWb9tg+A529sS7P6GucvInXTbT0aM7hDCEVWG/d/upmUrHyjI4nUSCr+ReqoY6dzeWDhFmw2GNE1lJs6NzI6kohIlTGZTEy/qR3N6ntyIiOfBz/bQpFV4/9FzqbiX6QOyi8sYvzHMaTlWGjX0IdnBmucv4jUfR4uTrz1r864mR3580Aqr/26z+hIIjWOin+ROujZ73ezLSEd379uhNM4fxGxF82DvHhxaDsAXv/9AMv3JhucSKRmUfEvUscsjkngk/XxmEwwa3hHQv3djY4kIlKtbujYkNt6hAEwcdFWjp3ONTiRSM2h4l+kDtl1PJ3Hvt4BwINXN+eKlvUNTiQiYownr2tN+0Y+nM6xMP7jGPIsRUZHEqkRVPyL1BHpORb+/fFm8gutXNEykAeuam50JBERw7g4OTL71s74uJnZlpDOU9/u1APARFDxL1InWK02HvpiK/Gncmjk58as4R1xcDAZHUtExFCh/u68MbITDib4fFMCH6+PNzqSiOFU/IvUAa//vp9fY5NxdnJg7m1R+Lo7Gx1JRKRG6NMikIevaQXAtO926QFgYvdU/IvUcj/vTGLWr/uB4gd5tW3oY3AiEZGa5d4+EVzXvgGFVhv//ngziem6AVjsl4p/kVpsb1ImD32+FYA7ejXhli6hxgYSEamBTCYTLw9rT6tgL1Ky8hn38WbdACx2S8W/SC11OqeAuz/cRHZBEb2a1uPxayONjiQiUmO5OzvxzqguxTcAHz2tG4DFbqn4F6mFCous3P/plpIbfN+8tTNmR/1zFhG5kLB6ugFYRNWCSC304k97WH0gBTezI++O7oK/h27wFREpj7NvAF57MNXgRCLVS8W/SC2zOCaB91bHAfDqLR2IbOBtcCIRkdrl3j4RXN8hpPgG4E9iOJKabXQkkWqj4l+kFtl69DRT/3qC7wNXNWNguwYGJxIRqX1MJhMzhrWnw19PAB6zYCPpuRajY4lUCxX/IrXEsdO53P3hJgoKrfSLDGJCvxZGRxIRqbVc/xo2GeztysGT2dz/6WYKi6xGxxKpcir+RWqBzDwLdy3YyMnMfFoFezFrhJ7gKyJyqep7u/Le7V1wMzuyan8Kz/8Ya3QkkSqn4l+khisssvKfhVvYk5RJoJcL8+7oiqeLk9GxRETqhLYNffjf8A4ALFhzmI/WHTE4kUjVUvEvUsM9/2MsK/aexNXswHuju9DQ183oSCIidco1bRswZUBLAJ75bher96cYnEik6qj4F6nBFvwZx4I1hwGYNbwjHUJ9Dc0jIlJXjb+iKUM6NaTIamP8JzEcSM4yOpJIlVDxL1JDLd+TzLM/7AbgkWtacU1bzewjIlJVTCYT029qR1RjPzLyCrlzwQZOZuYbHUuk0qn4F6mBYhMzuP/TzVhtcEuXRozrG2F0JBGROs/V7Mg7o6JoXM+do6dyGbNgI9n5hUbHEqlUKv5Fapjjp//6g1NQRM+Iejx/YztMJs3sIyJSHep5uvDBnd3w93Bmx7F0TQEqdY6Kf5EaJD3Hwh3vbyAxPY+mgR7MvS0KZyf9MxURqU5NAjx47/YuuJodWL73JE9+uwubzWZ0LJFKoapCpIbIsxRx90eb2HciiyBvFz4Y0w0fd7PRsURE7FLnMD9eH9EJBxMs3BDPnBUHjY4kUilU/IvUAEVWG5M+38qGuFN4uTix4M5uNPJzNzqWiIhdi24TzDOD2wAw45e9fLU5weBEIpdOxb+IwWw2G8/9sJslO5JwdnTg7dFRRDbwNjqWiIgAo3s24d6/Jl14+MvtegaA1Hoq/kUM9vYfh0rm8n/llg70ahpgbCARESnlkQGtuL5DCIVWG/d+tIltR08bHUnkoqn4FzHQlzEJvPjTHgCevK4113cIMTiRiIiczcHBxMyb23N5swCyC4q44/0NHEjONDqWyEVR8S9ikJ93JvLwl9sAuLt3OHddHm5wIhEROR8XJ0feHhVFh1Bf0nIs3PbeBhLScoyOJVJhKv5FDPDHvpP8Z+EWrDYY3iWUxwZFGh1JRET+gYeLEwvu6Erz+p4kZeQxap6eAiy1j4p/kWq26fAp7v0oBkuRjWvbNeCFm/QQLxGR2sLPw5mP7upOQ1834lKyuX3+BjLyLEbHEik3Ff8i1WjX8XTuXLCRXEsRfVsE8r/hHXF0UOEvIlKbBPu48vHY7gR4OrM7MYOxCzaRW1BkdCyRclHxL1JNDp7MYvS8DWTmFdK1iZ+e3isiUouFB3jwwZhueLk4seHwKe75aBN5Fr0BkJpPlYdINYhPzeG299aTml1A24bezLujK27OjkbHEhGRS9AmxIf37+yKu7Mjq/an8O+PY8gv1BsAqdlU/ItUsaOnchjxzloS0/NoVt+TD+7shrer2ehYIiJSCbo08Wf+HV1xNTuwfO9J7vtkCwWFVqNjiZyXin+RKlRc+K/jeHoeEYEefHp3d+p5uhgdS0REKlGPiHq8N7orzk4O/Bp7ggc/20Jhkd4ASM2k4l+kiiSk5TDy3XUcO51LRIAHn93dg/perkbHEhGRKnB58wDeGRWFs6MDP+1MYuLn2yiy2oyOJXIOFf8iVeD46VxGvruOhLRcmtRz59O7e1DfW4W/iEhddkXL+sz5V2ecHEx8v+04k7/Ypk8ApMZR8S9SyRLTcxnxzjqOnsqlcT13Ft7Tg2AfFf4iIvagX+sg3ry1E44OJr7ecowJi7Zi0RsAqUFU/ItUovjUHG6eu5b4UzmE+buz8O4eNPBxMzqWiIhUo2vaNmD2rZ0xO5r4YXsi932yWbMASY1xUcX/nDlzCA8Px9XVlaioKFatWnXB9itXriQqKgpXV1ciIiKYO3fuOW0WL15M69atcXFxoXXr1nz99del1j/zzDOYTKZSX8HBwRcTX6RKHEjO4pa315YM9Vl4Tw9CfFX4i4jYo2vaBvP2qOLnuSzdfYJ7P4rRcwCkRqhw8b9o0SImTJjA448/zpYtW+jduzcDBw4kPj6+zPZxcXEMGjSI3r17s2XLFh577DEeeOABFi9eXNJm7dq1DB8+nFGjRrFt2zZGjRrFLbfcwvr160vtq02bNiQmJpZ87dixo6LxRapEbGIGw99eS1JGHs3re/L5vT1pqMJfRMSuXdUqiHm3d8HV7MCKvSe564ON5BQUGh1L7FyFi/9XX32Vu+66i7FjxxIZGcmsWbMIDQ3lrbfeKrP93LlzCQsLY9asWURGRjJ27FjGjBnDzJkzS9rMmjWL/v37M3XqVFq1asXUqVO5+uqrmTVrVql9OTk5ERwcXPIVGBhY0fgilW7r0dOMeGcdqdkFtAnxZtG9PXVzr4iIANC7eSAL7uyGu7Mjfx5I5Y75G8nK1xsAMY5TRRoXFBQQExPDo48+Wmp5dHQ0a9asKXObtWvXEh0dXWrZgAEDmDdvHhaLBbPZzNq1a5k4ceI5bc4u/vfv309ISAguLi50796dF154gYiIiPPmzc/PJz8/v+T7jIwMACwWCxaL5R9/Xjm/M+fP3s9j8SPdt5BdUESnUB/eG9UZL2eTXZ4X9Qn5O/UHOZs994moUG/evz2Kuz7czIbDpxjx9lreG92Zeh7ORkczlD33icpWkXNYoeI/JSWFoqIigoKCSi0PCgoiKSmpzG2SkpLKbF9YWEhKSgoNGjQ4b5u/77N79+58+OGHtGjRghMnTvD888/Tq1cvdu3aRb169co89vTp05k2bdo5y5cuXYq7u3u5fma5sGXLlhkdwTA700ws2OeAxWqiubeVkQ1SWb3cfs/HGfbcJ+Rc6g9yNnvuE+NawFuxjuw8nsHgWcsZF1lEPX1QbNd9orLk5OSUu22Fiv8zTCZTqe9tNts5y/6p/dnL/2mfAwcOLPn/du3a0bNnT5o2bcoHH3zApEmTyjzu1KlTS63LyMggNDSU6OhovL29z5tX/pnFYmHZsmX0798fs9lsdJxq90VMAvPXx1JktdG3RQBvjuiAq9nR6FiGsvc+IaWpP8jZ1CeK9T+ZzZgPYzh2Oo+3D3gwb3RnWgZ7GR3LEOoTlefM6JbyqFDxHxAQgKOj4zlX+ZOTk8+5cn9GcHBwme2dnJxKrtifr8359gng4eFBu3bt2L9//3nbuLi44OLics5ys9msTlZJ7O1c2mw23vz9AK8s2wfAsKhGTL+pHWZHzZp7hr31Cbkw9Qc5m733iZYhviz+92WMnr+efSeyuHXeRubf0ZUuTfyNjmYYe+8TlaEi569CFYuzszNRUVHnfDyzbNkyevXqVeY2PXv2PKf90qVL6dKlS0nQ87U53z6heDx/bGwsDRo0qMiPIHLRiqw2nvp2V0nhP/6KpswY1l6Fv4iIVEiwjyuf39uTqMZ+ZOQV8q/31vNb7AmjY4mdqHDVMmnSJN577z3mz59PbGwsEydOJD4+nnHjxgHFQ21Gjx5d0n7cuHEcOXKESZMmERsby/z585k3bx6TJ08uafPggw+ydOlSXnrpJfbs2cNLL73Er7/+yoQJE0raTJ48mZUrVxIXF8f69esZNmwYGRkZ3H777Zfw44uUT56liPs+2cxH645gMsEz17fm4WtaXXC4m4iIyPn4ujvz8V3duapVffILrdzzUQwfrztidCyxAxUe8z98+HBSU1N59tlnSUxMpG3btixZsoTGjRsDkJiYWGrO//DwcJYsWcLEiROZPXs2ISEhvP766wwdOrSkTa9evfjss8944oknePLJJ2natCmLFi2ie/fuJW0SEhIYOXIkKSkpBAYG0qNHD9atW1dyXJGqciq7gHs/2sTGw2k4Ozrw6vAOXNc+xOhYIiJSy7k5O/L2qCge+2oHX8Qk8MQ3OzmSms3UgZE4OOjiklSNi7rhd/z48YwfP77MdQsWLDhnWd++fdm8efMF9zls2DCGDRt23vWfffZZhTKKVIYDyZmMWbCJ+FM5eLk48fboKHo1DTA6loiI1BFmRwdeHtaeJgEezPhlL++uiiP+VA6zhnfCzdm+J5KQqqHByiLnsWr/SYbMWUP8qRxC/d34anwvFf4iIlLpTCYT913ZjNdHdsLZyYFfdp1g+DtrSc7MMzqa1EEq/kXK8NG6I9zx/kYy8wrp2sSPb8ZfRvMg+5yKTUREqsfgDiF8OrY7fu5mtiekM2T2GvYklX8KR5HyUPEv8jeFRVamfb+LJ7/ZSZHVxk2dGvLx2O7U8zx3ylgREZHK1qWJP1+Pv4yIAA+Onc7lpjlr+GlHotGxpA5R8S/yl1PZBdz+/gbe//MwAFMGtOSVWzrg4qQxlyIiUn2aBHjw1fheXNasHjkFRfz7k83M/GUvRVab0dGkDlDxLwLsSEjn+jdW8+eBVNydHZnzr87cd2UzTeUpIiKG8HV35oM7u3F373AA3lx+gLs/3ER6rsXgZFLbqfgXu/dlTAJD567h2OlcmtRz5+vxlzGonR4eJyIixnJydODxa1sza3hHXJwc+H1PMkNm/8mB5Eyjo0ktpuJf7FZBoZWnvt3J5C+2UVBo5epW9fn2/stpGawbe0VEpOa4sVNDFv+7FyE+rhxKyebG2Wv4Yftxo2NJLaXiX+zSsdO5jHx3HR+uLX6a4oR+zXl3dBd83MwGJxMRETlX24Y+fPefy+ke7k9WfiH3f7qFJ7/ZSZ6lyOhoUsuo+Be788uuJAbO+oOYI2l4uTox7/YuTOjXQk9TFBGRGi3A04VPxnbnviubAsXTUg+bu4YjqdkGJ5PaRMW/2I38wiKe+W4X934UQ0ZeIR1CfVnyQG+ujgwyOpqIiEi5ODk6MGVAKxbc2RU/dzM7j2Vw3eurWaLpQKWcVPyLXTh0Moub5qxhwZrDANzTJ4Iv7u1JqL+7scFEREQuwhUt67Pkwd50aexHZn4h4z/ZzBPf7CC3QMOA5MJU/EudZrPZ+HzTUa5/YzW7jmfg527m/Tu68tigSJyd1P1FRKT2auDjxsJ7ejCub/EwoI/XxXPtG6vYkZBucDKpyVT9SJ11MjOfuz+M4eEvt5NdUES3cH9+erAPV7aqb3Q0ERGRSmF2dODRga346K5uBHm7cOhkNkPm/Mmbv++nsMhqdDypgVT8S530884kBsz6g19jT2B2NPHwNS1ZeHcPgn1cjY4mIiJS6Xo3D+SXCX0Y1C6YQquNmUv3MfyddcSn5hgdTWoYFf9Sp2TkWZj0+VbGfRzDqewCWgV78e19lzP+imY4ajYfERGpw3zdnZl9a2deubkDni5OxBxJY+Brf/DRuiNYrTaj40kNoeJf6oxfd59gwP/+4KvNx3Awwb+vaMq3919G6xBvo6OJiIhUC5PJxNCoRvz0YG+6NfEnu6CIJ7/ZyYh31nHoZJbR8aQGUPEvtV5yZh73fbqZsR9uIjE9j8b13Pn83p48ck0rXJwcjY4nIiJS7UL93fnsnh48c31r3J0d2XD4FANfW8XclQd1L4CdU/EvtZbNZmPRxnj6vbKSH7cn4uhg4t4+Efz8YB+6NPE3Op6IiIihHBxM3HFZOL9M6EPv5gHkF1p58ac9DJmzht3HM4yOJwZxMjqAyMU4dDKLx7/eydpDqQC0CfHmpaHtadvQx+BkIiIiNUuovzsfjunGlzEJPPfDbnYcS+f6N1dze88mTOjfHG9Xs9ERpRqp+JdaJSu/kDd+38/81XFYimy4mh2Y1L8FYy4Lx8lRH2SJiIiUxWQycXOXUPq2DOSZ73axZEcS8/+M4/vtx3l8UCQ3dAzBZNLEGPZAxb/UCjabjW+3HueFJbEkZ+YDcEXLQJ4d3JawenpKr4iISHnU93Jlzr+i+GPfSZ75bheHUrKZsGgrn26I57kb2tIy2MvoiFLFVPxLjbfreDrPfLeLjYfTAGhcz52nrmvN1ZFBBicTERGpnfq0COSnCb15b1Ucb/y+nw1xpxj0+ipu6x7GA1c3p56ni9ERpYqo+Jca6/jpXF5dto/FmxOw2cDN7Mj9VzXjrsvDcTVrFh8REZFL4eLkyH1XNuOGjiE898Nuftl1gg/WHuGrzccYf2Uz7rysif7e1kEq/qXGSc+1MGfFARb8eZj8wuLpyK5r34DHBkUS4utmcDoREZG6pZGfO2+P6sKaAyn8d0ksu45n8NLPe/h43RGmDGjJ4A4hOOhBmXWGin+pMfILi/ho7RHe+P0A6bkWALqH+zN1UCQdQ32NDSciIlLH9WoWwPf3X843W48x45e9HDudy4RFW3lv9SEe6t+SK1oG6qbgOkDFvxguv7CIzzclMGf5ARLT8wBoEeTJowNbcWXL+nqhERERqSYODiZu6tyIgW0bMP/POOYsP8DOYxncuWAjncN8mdS/JZc1q6e/zbWYin8xTFlFf7C3K5P6t2BoVCMc9RGjiIiIIdyci+8HGNE1lLf/OMSHaw+zOf40t81bT7dwfx7q34LuEfWMjikXQcW/VLs8SxFfxJxb9I+/sim3dAnVzUUiIiI1RD1PFx4bFMnYy8OZs+Ign66PZ0PcKYa/s47u4f78+4qm9G2h4UC1iYp/qTancwr4aO0RPlh7mJSsAkBFv4iISG1Q39uVZwa34d6+EcxefoBFG4+yPu4U6+NOEdnAm39f0ZRBbYP1wM1aQMW/VLmjp3KYtzqOzzcdJaegCICGvm7c0yeC4V1V9IuIiNQWDXzceP7Gdoy/ohnzVsexcEM8sYkZPLBwCzP93bm7TwRDOzfE3VklZk2l34xUCZvNxsbDaXy49jA/7UyiyGoDILKBN/f2ieDa9g0w6+qAiIhIrRTi68aT17XmP1c148O1R3j/zzjiT+Xw5Dc7mfnLXoZ3DWVUj8aE+rsbHVXOouJfKlV2fiFfbznGx+uOsCcps2R57+YB3NMngsubBWhcoIiISB3h6+7MA1c3Z2zvcBZtPMr7fx4m/lQO7/xxiPdWHaJfZBB3XNaEnhGaIaimUPEvleJAciYfrT3C4s3HyMovBIqfyHtjpxBG9WhC6xBvgxOKiIhIVXF3duLOy8IZ3bMJK/Yms2DNYVbtT2Hp7hMs3X2CpoEejOgaxpDODQnwdDE6rl1T8S8XLacQFm48ytdbE9kSf7pkeUSAB7f1aMzQqEb4uJmNCygiIiLVytHBxNWRQVwdGcT+E5l8sPYwX20+xsGT2fx3SSwv/byH/q2DuKVrKD2b+Bod1y6p+JcKKbLaWH0ghc83xvPLTkcKbbFA8T/2q1rVZ3TPxlzWNECPARcREbFzzYO8eP7GdjxyTSu+35bIoo3xbEtI56edSfy0M4lgbxfaeznQIjmLyIZ+Rse1Gyr+5R/ZbDa2Hj3Nj9sT+WF7IkkZeX+tMdGivic3dwnlhk4h1PdyNTSniIiI1DxermZu7R7Grd3D2JOUwaKNR/l6yzGSMvJJynBg6RtriGzgzQ0dQ7i+QwgNfd2MjlynqfiXMtlsNnYcS+eH7Yn8uD2RY6dzS9b5upu5vn0DgrIPcc/NPXF2djYwqYiIiNQWrYK9efr6Njw6sBU/bT/Oe8u2sjfDkdjEDGITM3jxpz10aezH4I4hDGgTTJC3LixWNhX/UqKwyMqmI2n8FnuCn3clcfTU/xf8Hs6O9GsdxKB2DbiiZSAONitLlhzSnfsiIiJSYS5OjlzbLhjTUSu9rriaX/em8N3W46yLS2XTkTQ2HUnjqW930THUl/6tgxjQJoimgZ6qOyqBin87l55rYeW+k/wWe4IVe0+SnmspWedmduTqyPpc174BV7SsX+phXBaL1Yi4IiIiUsf4upsZ2S2Mkd3CSErP44ftx/lxR/FkIluPFn/N+GUvEQEe9G8TxJUt6xPV2E/PC7pIKv7tTGGRlR3H0vnzQAqr9qcQcySNwr8ewAXg527mypb16de6+B+Xm7OevisiIiLVI9jHlbG9IxjbO4LkjDx+jU1m6e4k1hxI5VBKNm+vPMTbKw/h4exIz6YB9G0RQJ8WgTSu52F09FpDxX8dZ7PZiEvJLin21x5KJTOvsFSb5vU9uToyiH6R9ekU5oejZuoRERERg9X3di25UTgzr3ikwrLdJ1i1P4VT2QX8GnuCX2NPANC4njt9mgfSq2k9uob761kCF6Div44pLLKyJymTTYdPsfFIGjGH0/42O08xb1cnejUN4LLmAfRpHqB3yyIiIlKjebmaua59CNe1D8FqtbHreAZ/7D/JH/tOEnMkjSOpOXyUeoSP1h0BoGmgB93C69E93J9u4f6EaAahEir+a7nTOQXsOJZOzJE0Yo6ksflIGtkFRaXaODs60KWJH5c1C+DyZgG0beijq/siIiJSKzk4mGjXyId2jXy478pmZOUXsvZgKqv3n2R93Cn2JGVy8GQ2B09ms3BDPACN/NzoHOZHh1BfOob60CbEp9S9jPZExX8tkp5jYcex9L++TrPjWHqpGXnO8HJxonNjP7o28aNLE386NPLV2H0RERGpkzxdnOjfOoj+rYOA4gujGw+nsSEulQ1xp9h5PIOEtFwS0nL5bttxoPjhpC2DvEreDLRv5EvTQE+cner+TcQq/mugwiIrh1Oz2ZuUxd4TmexLymR3Ygbxp3LKbN+knjsdQn3p0sSfLo39aBHkpSv7IiIiYpd83Z1LvRnIyi9kS3wa246eZuvRdLYePU1KVj67EzPYnZjBwg3F25kdTTQN9CSygTetgr1o1cCbyGAvAr1c6tQUoyr+DZRbUMTh1GwOp2RzKCWb/Scy2Xsii4PJWRQUlT2VZpi/O+0aFn/U1b6hD20a+uDjZq7m5CIiIiK1g6eLE72bB9K7eSBQPBlKYnoe2xOK3wxsO3qancfTycwrZE9SJnuSMktt7+/hTPP6nkQEetI00IOIQA8iAjxp5OeGUy2cblTFfxXLzLNw7HQuR0/llhT5h1OyOZyaTWJ63nm3c3d2pHmQFy2DPGkR5EWrYG/aNvTG111P0xURERG5WCaTiRBfN0J83bimbQOg+A3B8fQ89vz1pOHYpEz2JGYQl5LNqewC1sedYn3cqVL7cXZ0oHE9dyICPQgP8CTU341Gfu6E+hXvu6beU6Di/xIUFllJySogKSOPY2m5HDud89d/czl2Oo9jaTlknDWt5tm8XZ0ID/QkvJ77X8W+Fy2DvWjo64aDhu6IiIiIVDmTyURDXzca+rpxdWRQyfI8SxH7T2Rx4GQmh05mc+hkNgdPZhGXkk1+oZX9yVnsT84CTpyzz/peLoT6u9PIz+2vL3eCvV2p7+1CkLcr/u7OhtR6Kv7PYrXayMwr5FROASlZ+SRn5JOcmUdyZj4nM/NJzswnOSOPlKx8UrMLsNn+eZ++7mYa+rrRJMCDiAAPmtTzoEmAB+EBHvi5m+vUODIRERGRusLV7Fgys9DfWa02jp3O5VBKNodOZnE4JbtkpMfRtBxyCoqKa8bMfGKOpJW5bycHE/W9XKjv7UrQX28IgrxdCfR0wd/DGX9PZ+p5OOPn4YyXi1Ol1YsXVfzPmTOHGTNmkJiYSJs2bZg1axa9e/c+b/uVK1cyadIkdu3aRUhICA8//DDjxo0r1Wbx4sU8+eSTHDx4kKZNm/Lf//6XIUOGXNJxz+fdVQfJN7mSll1AWo6FtJwC0nIKOJ1j4XROAdZyFPRnODqYCPR0oaFf8bvFhn991NPob//v6aL3WCIiIiJ1hYODiVB/d0L93enbIrDUOpvNxukcCwlpxW8EEtJySEjL5VhaLicy8ziRkU9KVj6F1uKhRscvMAz8DGdHB/w8zPi5O1PP0xl/Dxf83c34uDvj62bG2frP+zijwlXpokWLmDBhAnPmzOGyyy7j7bffZuDAgezevZuwsLBz2sfFxTFo0CDuvvtuPv74Y/7880/Gjx9PYGAgQ4cOBWDt2rUMHz6c5557jiFDhvD1119zyy23sHr1arp3735Rx72Q1349gIOL+wXbuDs7EuDp8tc7Mhfqe7kS6OVCoNdfy7yKP7Yx6iMbEREREal5TCYTfn9dsT/7E4Mz/j50/ERGHskZxW8KTmTkcTIrn1PZBaRmFV+czikooqDI+tf6/DL3Z80ve0bIMvPZbOUZuPL/unfvTufOnXnrrbdKlkVGRnLjjTcyffr0c9o/8sgjfPfdd8TGxpYsGzduHNu2bWPt2rUADB8+nIyMDH766aeSNtdccw1+fn4sXLjwoo5bloyMDHx8fLjv/dUEB/jh5+GMr7sZf3dnfN2dS95R+bqbcXGqmTdp1BQWi4UlS5YwaNAgzGbNNiTqE1Ka+oOcTX1CzqY+UT55liJSswtIyy4gNbuAU9n5JW8MTudYSM+1cPLUaT7/z9Wkp6fj7e19wf1V6Mp/QUEBMTExPProo6WWR0dHs2bNmjK3Wbt2LdHR0aWWDRgwgHnz5mGxWDCbzaxdu5aJEyee02bWrFkXfVyA/Px88vP//x1SRkYGANOub3XhE2OzYrGUPdWmFLNYLKX+K6I+IX+n/iBnU5+Qs6lPlI8jUN/DifoeTkDZI1cyMjL4/D/l21+Fiv+UlBSKiooICgoqtTwoKIikpKQyt0lKSiqzfWFhISkpKTRo0OC8bc7s82KOCzB9+nSmTZt2zvKlS5fi7n7hYT9SPsuWLTM6gtQw6hPyd+oPcjb1CTmb+sSly8kp/7Cfi7oT9ey7jW022wXvQC6r/dnLy7PPih536tSpTJo0qeT7jIwMQkNDiY6O/sePROTCLBYLy5Yto3///vqoTgD1CSlN/UHOpj4hZ1OfqDxnRreUR4WK/4CAABwdHc+52p6cnHzOVfkzgoODy2zv5OREvXr1LtjmzD4v5rgALi4uuLi4nLPcbDark1USnUs5m/qE/J36g5xNfULOpj5x6Spy/ir0TGJnZ2eioqLO+Xhm2bJl9OrVq8xtevbseU77pUuX0qVLl5Kg52tzZp8Xc1wRERERESmtwsN+Jk2axKhRo+jSpQs9e/bknXfeIT4+vmTe/qlTp3Ls2DE+/PBDoHhmnzfffJNJkyZx9913s3btWubNm1cyiw/Agw8+SJ8+fXjppZe44YYb+Pbbb/n1119ZvXp1uY8rIiIiIiIXVuHif/jw4aSmpvLss8+SmJhI27ZtWbJkCY0bNwYgMTGR+Pj4kvbh4eEsWbKEiRMnMnv2bEJCQnj99ddL5vgH6NWrF5999hlPPPEETz75JE2bNmXRokUlc/yX57giIiIiInJhF3XD7/jx4xk/fnyZ6xYsWHDOsr59+7J58+YL7nPYsGEMGzbsoo8rIiIiIiIXVqEx/yIiIiIiUnup+BcRERERsRMq/kVERERE7ISKfxERERERO3FRN/zWVmeeLFyRp6BJ2SwWCzk5OWRkZOjBHAKoT0hp6g9yNvUJOZv6ROU5U9ueqXUvxK6K/8zMTABCQ0MNTiIiIiIiUrkyMzPx8fG5YBuTrTxvEeoIq9XK8ePH8fLywmQyGR2nVsvIyCA0NJSjR4/i7e1tdBypAdQn5O/UH+Rs6hNyNvWJymOz2cjMzCQkJAQHhwuP6rerK/8ODg40atTI6Bh1ire3t/7BSinqE/J36g9yNvUJOZv6ROX4pyv+Z+iGXxERERERO6HiX0RERETETqj4l4vi4uLC008/jYuLi9FRpIZQn5C/U3+Qs6lPyNnUJ4xhVzf8ioiIiIjYM135FxERERGxEyr+RURERETshIp/ERERERE7oeJfRERERMROqPiX8yosLOSJJ54gPDwcNzc3IiIiePbZZ7FarSVtbDYbzzzzDCEhIbi5uXHFFVewa9cuA1NLZfrjjz+4/vrrCQkJwWQy8c0335RaX57ff35+Pv/5z38ICAjAw8ODwYMHk5CQUI0/hVSmC/UJi8XCI488Qrt27fDw8CAkJITRo0dz/PjxUvtQn6g7/uk14u/uvfdeTCYTs2bNKrVc/aFuKU+fiI2NZfDgwfj4+ODl5UWPHj2Ij48vWa8+UbVU/Mt5vfTSS8ydO5c333yT2NhYXn75ZWbMmMEbb7xR0ubll1/m1Vdf5c0332Tjxo0EBwfTv39/MjMzDUwulSU7O5sOHTrw5ptvlrm+PL//CRMm8PXXX/PZZ5+xevVqsrKyuO666ygqKqquH0Mq0YX6RE5ODps3b+bJJ59k8+bNfPXVV+zbt4/BgweXaqc+UXf802vEGd988w3r168nJCTknHXqD3XLP/WJgwcPcvnll9OqVStWrFjBtm3bePLJJ3F1dS1poz5RxWwi53HttdfaxowZU2rZTTfdZLvttttsNpvNZrVabcHBwbYXX3yxZH1eXp7Nx8fHNnfu3GrNKlUPsH399dcl35fn93/69Gmb2Wy2ffbZZyVtjh07ZnNwcLD9/PPP1ZZdqsbZfaIsGzZssAG2I0eO2Gw29Ym67Hz9ISEhwdawYUPbzp07bY0bN7b973//K1mn/lC3ldUnhg8fXlJHlEV9ourpyr+c1+WXX85vv/3Gvn37ANi2bRurV69m0KBBAMTFxZGUlER0dHTJNi4uLvTt25c1a9YYklmqT3l+/zExMVgsllJtQkJCaNu2rfqInUhPT8dkMuHr6wuoT9gbq9XKqFGjmDJlCm3atDlnvfqDfbFarfz444+0aNGCAQMGUL9+fbp3715qaJD6RNVT8S/n9cgjjzBy5EhatWqF2WymU6dOTJgwgZEjRwKQlJQEQFBQUKntgoKCStZJ3VWe339SUhLOzs74+fmdt43UXXl5eTz66KPceuuteHt7A+oT9uall17CycmJBx54oMz16g/2JTk5maysLF588UWuueYali5dypAhQ7jppptYuXIloD5RHZyMDiA116JFi/j444/59NNPadOmDVu3bmXChAmEhIRw++23l7QzmUyltrPZbOcsk7rrYn7/6iN1n8ViYcSIEVitVubMmfOP7dUn6p6YmBhee+01Nm/eXOHfrfpD3XRmwpAbbriBiRMnAtCxY0fWrFnD3Llz6du373m3VZ+oPLryL+c1ZcoUHn30UUaMGEG7du0YNWoUEydOZPr06QAEBwcDnPNOPDk5+ZyrwVL3lOf3HxwcTEFBAWlpaedtI3WPxWLhlltuIS4ujmXLlpVc9Qf1CXuyatUqkpOTCQsLw8nJCScnJ44cOcJDDz1EkyZNAPUHexMQEICTkxOtW7cutTwyMrJkth/1iaqn4l/OKycnBweH0l3E0dGx5J17eHg4wcHBLFu2rGR9QUEBK1eupFevXtWaVapfeX7/UVFRmM3mUm0SExPZuXOn+kgddabw379/P7/++iv16tUrtV59wn6MGjWK7du3s3Xr1pKvkJAQpkyZwi+//AKoP9gbZ2dnunbtyt69e0st37dvH40bNwbUJ6qDhv3IeV1//fX897//JSwsjDZt2rBlyxZeffVVxowZAxQP95gwYQIvvPACzZs3p3nz5rzwwgu4u7tz6623GpxeKkNWVhYHDhwo+T4uLo6tW7fi7+9PWFjYP/7+fXx8uOuuu3jooYeoV68e/v7+TJ48mXbt2tGvXz+jfiy5BBfqEyEhIQwbNozNmzfzww8/UFRUVPLJkL+/P87OzuoTdcw/vUac/ebPbDYTHBxMy5YtAb1G1EX/1CemTJnC8OHD6dOnD1deeSU///wz33//PStWrADUJ6qFkVMNSc2WkZFhe/DBB21hYWE2V1dXW0REhO3xxx+35efnl7SxWq22p59+2hYcHGxzcXGx9enTx7Zjxw4DU0tlWr58uQ045+v222+32Wzl+/3n5uba7r//fpu/v7/Nzc3Ndt1119ni4+MN+GmkMlyoT8TFxZW5DrAtX768ZB/qE3XHP71GnO3sqT5tNvWHuqY8fWLevHm2Zs2a2VxdXW0dOnSwffPNN6X2oT5RtUw2m81WXW80RERERETEOBrzLyIiIiJiJ1T8i4iIiIjYCRX/IiIiIiJ2QsW/iIiIiIidUPEvIiIiImInVPyLiIiIiNgJFf8iIiIiInZCxb+IiFSJK664ggkTJpRatnfvXoKDg8nMzARgwYIF+Pr6XtJxyrOPyZMn88ADD1zScURE6gIV/yIiUm0ef/xx7rvvPry8vKr1uA8//DDvv/8+cXFx1XpcEZGaRsW/iIhUi4SEBL777jvuvPPOaj92/fr1iY6OZu7cudV+bBGRmkTFv4hIHfDll1/Srl073NzcqFevHv369SM7O5s77riDG2+8kWnTplG/fn28vb259957KSgoKNnWZrPx8ssvExERgZubGx06dODLL78stf/du3czaNAgPD09CQoKYtSoUaSkpJSsz87OZvTo0Xh6etKgQQNeeeWVczJ+/vnndOjQgUaNGp3350hNTaVbt24MHjyYvLw8VqxYgclk4scff6RDhw64urrSvXt3duzYcc62v/zyC5GRkXh6enLNNdeQmJhYav3gwYNZuHBhuc+piEhdpOJfRKSWS0xMZOTIkYwZM4bY2FhWrFjBTTfdhM1mA+C3334jNjaW5cuXs3DhQr7++mumTZtWsv0TTzzB+++/z1tvvcWuXbuYOHEit912GytXrizZf9++fenYsSObNm3i559/5sSJE9xyyy0l+5gyZQrLly/n66+/ZunSpaxYsYKYmJhSOf/44w+6dOly3p8jISGB3r1706pVK7766itcXV1L7X/mzJls3LiR+vXrM3jwYCwWS8n6nJwcZs6cyUcffcQff/xBfHw8kydPLrX/bt26cfToUY4cOXIRZ1lEpI6wiYhIrRYTE2MDbIcPHz5n3e23327z9/e3ZWdnlyx76623bJ6enraioiJbVlaWzdXV1bZmzZpS29111122kSNH2mw2m+3JJ5+0RUdHl1p/9OhRG2Dbu3evLTMz0+bs7Gz77LPPStanpqba3NzcbA8++GDJsg4dOtieffbZUvt5//33bT4+Pra9e/fawsLCbP/5z39sVqu1ZP3y5cttQJn7XrRoUck+ANuBAwdK2syePdsWFBRU6ljp6ek2wLZixYqyT6SIiB1wMvSdh4iIXLIOHTpw9dVX065dOwYMGEB0dDTDhg3Dz8+vZL27u3tJ+549e5KVlcXRo0dJTk4mLy+P/v37l9pnQUEBnTp1AiAmJobly5fj6el5zrEPHjxIbm4uBQUF9OzZs2S5v78/LVu2LNU2Nze31NX8vy+//PLLGTlyJK+99lqZP2NZ+46NjS1Z5u7uTtOmTUu+b9CgAcnJyaX24ebmBhR/SiAiYq9U/IuI1HKOjo4sW7aMNWvWsHTpUt544w0ef/xx1q9ff8HtTCYTVqsVgB9//JGGDRuWWu/i4gKA1Wrl+uuv56WXXjpnHw0aNGD//v3lyhkQEEBaWto5y11cXOjXrx8//vgjU6ZMueA9AWfnP8NsNp+zzvbXsKczTp06BUBgYGC59i8iUhdpzL+ISB1gMpm47LLLmDZtGlu2bMHZ2Zmvv/4agG3btpGbm1vSdt26dXh6etKoUSNat26Ni4sL8fHxNGvWrNRXaGgoAJ07d2bXrl00adLknDYeHh40a9YMs9nMunXrSo6RlpbGvn37SmXs1KkTu3fvPie7g4MDH330EVFRUVx11VUcP378nDZl7btVq1YVOkc7d+7EbDbTpk2bCm0nIlKXqPgXEanl1q9fzwsvvMCmTZuIj4/nq6++4uTJk0RGRgLFQ3juuusudu/ezU8//cTTTz/N/fffj4ODA15eXkyePJmJEyfywQcfcPDgQbZs2cLs2bP54IMPALjvvvs4deoUI0eOZMOGDRw6dIilS5cyZswYioqK8PT05K677mLKlCn89ttv7Ny5kzvuuAMHh9J/YgYMGMDatWspKio652dwdHTkk08+oUOHDlx11VUkJSWVWv/ss8+W2ndAQAA33nhjhc7TqlWr6N27d8nwHxERe6RhPyIitZy3tzd//PEHs2bNIiMjg8aNG/PKK68wcOBAFi1axNVXX03z5s3p06cP+fn5jBgxgmeeeaZk++eee4769eszffp0Dh06hK+vL507d+axxx4DICQkhD///JNHHnmEAQMGkJ+fT+PGjbnmmmtKCvwZM2aQlZXF4MGD8fLy4qGHHiI9Pb1UzkGDBmE2m/n1118ZMGDAOT+Hk5MTCxcuZPjw4Vx11VWsWLGiZN2LL77Igw8+yP79++nQoQPfffcdzs7OFTpPCxcuLDXLkYiIPTLZzh4UKSIidcYdd9zB6dOn+eabb4yOAsCcOXP49ttv+eWXX8rVfsWKFVx55ZWkpaXh6+t70cc9cz/B9u3bcXLSdS8RsV96BRQRkWpzzz33kJaWRmZmJl5eXtV23OzsbN5//30V/iJi9/QqKCIi1cbJyYnHH3+82o/79weSiYjYMw37ERERERGxE5rtR0RERETETqj4FxERERGxEyr+RURERETshIp/ERERERE7oeJfRERERMROqPgXEREREbETKv5FREREROyEin8RERERETuh4l9ERERExE78Hx2DIGeN09/MAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_gaussian_pdf(mean=120, variance=17**2, xlabel='speed(kph)');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The y-axis depicts the *probability density* — the relative amount of cars that are going the speed at the corresponding x-axis. I will explain this further in the next section.\n",
    "\n",
    "The Gaussian model is imperfect. Though these charts do not show it, the *tails* of the distribution extend out to infinity. *Tails* are the far ends of the curve where the values are the lowest. Of course human heights or automobile speeds cannot be less than zero, let alone $-\\infty$ or $\\infty$. “The map is not the territory” is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. Gaussians are used in many branches of mathematics, not because they perfectly model reality, but because they are easier to use than any other relatively accurate choice. However, even in this book Gaussians will fail to model reality, forcing us to use computationally expensive alternatives. \n",
    "\n",
    "You will hear these distributions called *Gaussian distributions* or *normal distributions*.  *Gaussian* and *normal* both mean the same thing in this context, and are used interchangeably. I will use both throughout this book as different sources will use either term, and I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and talk about a *Gaussian* or *normal* — these are both typical shortcut names for the *Gaussian distribution*. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gaussian Distributions\n",
    "\n",
    "Let's explore how Gaussians work. A Gaussian is a *continuous probability distribution* that is completely described with two parameters, the mean ($\\mu$) and the variance ($\\sigma^2$). It is defined as:\n",
    "\n",
    "$$ \n",
    "f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\big [{-\\frac{(x-\\mu)^2}{2\\sigma^2} }\\big ]\n",
    "$$\n",
    "\n",
    "$\\exp[x]$ is notation for $e^x$.\n",
    "\n",
    "<p> Don't be dissuaded by the equation if you haven't seen it before; you will not need to memorize or manipulate it. The computation of this function is stored in `stats.py` with the function `gaussian(x, mean, var, normed=True)`. \n",
    "    \n",
    "Shorn of the constants, you can see it is a simple exponential:\n",
    "    \n",
    "$$f(x)\\propto e^{-x^2}$$\n",
    "\n",
    "which has the familiar bell curve shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(-3, 3, .01)\n",
    "plt.plot(x, np.exp(-x**2));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's remind ourselves how to look at the code for a function. In a cell, type the function name followed by two question marks and press CTRL+ENTER. This will open a popup window displaying the source. Uncomment the next cell and try it now."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "from filterpy.stats import gaussian\n",
    "#gaussian??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot a Gaussian with a mean of 22 $(\\mu=22)$, with a variance of 4 $(\\sigma^2=4)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_gaussian_pdf(22, 4, mean_line=True, xlabel='$^{\\circ}C$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What does this curve *mean*? Assume we have a thermometer which reads 22°C. No thermometer is perfectly accurate, and so we expect that each reading will be slightly off the actual value. However, a theorem called  [*Central Limit Theorem*](https://en.wikipedia.org/wiki/Central_limit_theorem) states that if we make many measurements that the measurements will be normally distributed. When we look at this chart we can see it is proportional to the probability of the thermometer reading a particular value given the actual temperature of 22°C. \n",
    "\n",
    "Recall that a Gaussian distribution is *continuous*. Think of an infinitely long straight line - what is the probability that a point you pick randomly is at 2. Clearly 0%, as there is an infinite number of choices to choose from. The same is true for normal distributions; in the graph above the probability of being *exactly* 2°C is 0% because there are an infinite number of values the reading can take.\n",
    "\n",
    "What is this curve? It is something we call the *probability density function.* The area under the curve at any region gives you the probability of those values. So, for example, if you compute the area under the curve between 20 and 22 the resulting area will be the probability of the temperature reading being between those two temperatures. \n",
    "\n",
    "Here is another way to understand it. What is the *density* of a rock, or a sponge? It is a measure of how much mass is compacted into a given space. Rocks are dense, sponges less so. So, if you wanted to know how much a rock weighed but didn't have a scale, you could take its volume and multiply by its density. This would give you its mass. In practice density varies in most objects, so you would integrate the local density across the rock's volume.\n",
    "\n",
    "$$M = \\iiint_R p(x,y,z)\\, dV$$\n",
    "\n",
    "We do the same with *probability density*. If you want to know the temperature being between 20°C and 21°C you would integrate the curve above from 20 to 21. As you know the integral of a curve gives you the area under the curve. Since this is a curve of the probability density, the integral of the density is the probability. \n",
    "\n",
    "What is the probability of the temperature being exactly 22°C? Intuitively, 0. These are real numbers, and the odds of 22°C vs, say, 22.00000000000017°C is infinitesimal. Mathematically, what would we get if we integrate from 22 to 22? Zero. \n",
    "\n",
    "Thinking back to the rock, what is the weight of an single point on the rock? An infinitesimal point must have no weight. It makes no sense to ask the weight of a single point, and it makes no sense to ask about the probability of a continuous distribution having a single value. The answer for both is obviously zero.\n",
    "\n",
    "In practice our sensors do not have infinite precision, so a reading of 22°C implies a range, such as 22 $\\pm$ 0.1°C, and we can compute the probability of that range by integrating from 21.9 to 22.1.\n",
    "\n",
    "We can think of this in Bayesian terms or frequentist terms. As a Bayesian, if the thermometer reads exactly 22°C, then our belief is described by the curve - our belief that the actual (system) temperature is near 22°C is very high, and our belief that the actual temperature is near 18 is very low. As a frequentist we would say that if we took 1 billion temperature measurements of a system at exactly 22°C, then a histogram of the measurements would look like this curve. \n",
    "\n",
    "How do you compute the probability, or area under the curve? You integrate the equation for the Gaussian \n",
    "\n",
    "$$ \\int^{x_1}_{x_0}  \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}{(x-\\mu)^2}/\\sigma^2 } dx$$\n",
    "\n",
    "This is called the *cumulative probability distribution*, commonly abbreviated *cdf*.\n",
    "\n",
    "I wrote `filterpy.stats.norm_cdf` which computes the integral for you. For example, we can compute"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cumulative probability of range 21.5 to 22.5 is 19.74%\n",
      "Cumulative probability of range 23.5 to 24.5 is 12.10%\n"
     ]
    }
   ],
   "source": [
    "from filterpy.stats import norm_cdf\n",
    "print('Cumulative probability of range 21.5 to 22.5 is {:.2f}%'.format(\n",
    "      norm_cdf((21.5, 22.5), 22,4)*100))\n",
    "print('Cumulative probability of range 23.5 to 24.5 is {:.2f}%'.format(\n",
    "      norm_cdf((23.5, 24.5), 22,4)*100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean ($\\mu$) is what it sounds like — the average of all possible probabilities. Because of the symmetric shape of the curve it is also the tallest part of the curve. The thermometer reads 22°C, so that is what we used for the mean.  \n",
    "\n",
    "The notation for a normal distribution for a random variable $X$ is $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$ where $\\sim$ means *distributed according to*. This means I can express the temperature reading of our thermometer as\n",
    "\n",
    "$$\\text{temp} \\sim \\mathcal{N}(22,4)$$\n",
    "\n",
    "This is an extremely important result. Gaussians allow me to capture an infinite number of possible values with only two numbers! With the values $\\mu=22$ and $\\sigma^2=4$ I can compute the distribution of measurements over any range.\n",
    "\n",
    "Some sources use $\\mathcal N (\\mu, \\sigma)$ instead of $\\mathcal N (\\mu, \\sigma^2)$. Either is fine, they are both conventions. You need to keep in mind which form is being used if you see a term such as $\\mathcal{N}(22,4)$. In this book I always use $\\mathcal N (\\mu, \\sigma^2)$, so $\\sigma=2$, $\\sigma^2=4$ for this example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Variance and Belief\n",
    "\n",
    "Since this is a probability density distribution it is required that the area under the curve always equals one. This should be intuitively clear — the area under the curve represents all possible outcomes, *something* happened, and the probability of *something happening* is one, so the density must sum to one. We can prove this ourselves with a bit of code. (If you are mathematically inclined, integrate the Gaussian equation from $-\\infty$ to $\\infty$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0\n"
     ]
    }
   ],
   "source": [
    "print(norm_cdf((-1e8, 1e8), mu=0, var=4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This leads to an important insight. If the variance is small the curve will be narrow. this is because the variance is a measure of *how much* the samples vary from the mean. To keep the area equal to 1, the curve must also be tall. On the other hand if the variance is large the curve will be wide, and thus it will also have to be short to make the area equal to 1.\n",
    "\n",
    "Let's look at that graphically. We will use the aforementioned `filterpy.stats.gaussian` which can take either a single value or array of values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.24197072451914337\n",
      "[0.378 0.622]\n"
     ]
    }
   ],
   "source": [
    "from filterpy.stats import gaussian\n",
    "\n",
    "print(gaussian(x=3.0, mean=2.0, var=1))\n",
    "print(gaussian(x=[3.0, 2.0], mean=2.0, var=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default `gaussian` normalizes the output, which turns the output back into a probability distribution. Use the argument`normed` to control this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.242 0.399]\n"
     ]
    }
   ],
   "source": [
    "print(gaussian(x=[3.0, 2.0], mean=2.0, var=1, normed=False))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the Gaussian is not normalized it is called a *Gaussian function* instead of *Gaussian distribution*.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = np.arange(15, 30, 0.05)\n",
    "plt.plot(xs, gaussian(xs, 23, 0.2**2), label='$\\sigma^2=0.2^2$')\n",
    "plt.plot(xs, gaussian(xs, 23, .5**2), label='$\\sigma^2=0.5^2$', ls=':')\n",
    "plt.plot(xs, gaussian(xs, 23, 1**2), label='$\\sigma^2=1^2$', ls='--')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is this telling us? The Gaussian with $\\sigma^2=0.2^2$ is very narrow. It is saying that we believe $x=23$, and that we are very sure about that: within $\\pm 0.2$ std. In contrast, the Gaussian with $\\sigma^2=1^2$ also believes that $x=23$, but we are much less sure about that. Our belief that $x=23$ is lower, and so our belief about the likely possible values for $x$ is spread out — we think it is quite likely that $x=20$ or $x=26$, for example. $\\sigma^2=0.2^2$ has almost completely eliminated $22$ or $24$ as possible values, whereas $\\sigma^2=1^2$ considers them nearly as likely as $23$.\n",
    "\n",
    "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The curve for $\\sigma^2=0.2^2$ represents a very accurate thermometer, and curve for $\\sigma^2=1^2$ represents a fairly inaccurate one. Note the very powerful property the Gaussian distribution affords us — we can entirely represent both the reading and the error of a thermometer with only two numbers — the mean and the variance.\n",
    "\n",
    "An equivalent formation for a Gaussian is $\\mathcal{N}(\\mu,1/\\tau)$ where $\\mu$ is the *mean* and $\\tau$ the *precision*. $1/\\tau = \\sigma^2$; it is the reciprocal of the variance. While we do not use this formulation in this book, it underscores that the variance is a measure of how precise our data is. A small variance yields large precision — our measurement is very precise. Conversely, a large variance yields low precision — our belief is spread out across a large area. You should become comfortable with thinking about Gaussians in these equivalent forms. In Bayesian terms Gaussians reflect our *belief* about a measurement, they express the *precision* of the measurement, and they express how much *variance* there is in the measurements. These are all different ways of stating the same fact.\n",
    "\n",
    "I'm getting ahead of myself, but in the next chapters we will use Gaussians to express our belief in things like the estimated position of the object we are tracking, or the accuracy of the sensors we are using."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The  68-95-99.7 Rule\n",
    "\n",
    "It is worth spending a few words on standard deviation now. The standard deviation is a measure of how much the data deviates from the mean. For Gaussian distributions, 68% of all the data falls within one standard deviation ($\\pm1\\sigma$) of the mean, 95% falls within two standard deviations ($\\pm2\\sigma$), and 99.7% within three ($\\pm3\\sigma$). This is often called the [68-95-99.7 rule](https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule). If you were told that the average test score in a class was 71 with a standard deviation of 9.4, you could conclude that 95% of the students received a score between 52.2 and 89.8 if the distribution is normal (that is calculated with $71 \\pm (2 * 9.4)$). \n",
    "\n",
    "Finally, these are not arbitrary numbers. If the Gaussian for our position is $\\mu=22$ meters, then the standard deviation also has units meters. Thus $\\sigma=0.2$ implies that 68% of the measurements range from 21.8 to 22.2 meters. Variance is the standard deviation squared, thus $\\sigma^2 = .04$ meters$^2$. As you saw in the last section, writing $\\sigma^2 = 0.2^2$ can make this somewhat more meaningful, since the 0.2 is in the same units as the data.\n",
    "\n",
    "The following graph depicts the relationship between the standard deviation and the normal distribution. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from kf_book.gaussian_internal import display_stddev_plot\n",
    "display_stddev_plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interactive Gaussians\n",
    "\n",
    "For those that are reading this in a Jupyter Notebook, here is an interactive version of the Gaussian plots. Use the sliders to modify $\\mu$ and $\\sigma^2$. Adjusting $\\mu$ will move the graph to the left and right because you are adjusting the mean, and adjusting $\\sigma^2$ will make the bell curve thicker and thinner."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "69760c5aa4694c629a52688ae1706038",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=5.0, description='mu', max=7.0, min=3.0), FloatSlider(value=0.03, desc…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import math\n",
    "from ipywidgets import interact, FloatSlider\n",
    "\n",
    "def plt_g(mu,variance):\n",
    "    plt.figure()\n",
    "    xs = np.arange(2, 8, 0.01)\n",
    "    ys = gaussian(xs, mu, variance)\n",
    "    plt.plot(xs, ys)\n",
    "    plt.ylim(0, 0.04)\n",
    "    plt.show()\n",
    "\n",
    "interact(plt_g, mu=FloatSlider(value=5, min=3, max=7),\n",
    "         variance=FloatSlider(value = .03, min=.01, max=1.));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, if you are reading this online, here is an animation of a Gaussian. First, the mean is shifted to the right. Then the mean is centered at $\\mu=5$ and the variance is modified.\n",
    "\n",
    "<img src='animations/04_gaussian_animate.gif'>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computational Properties of Normally Distributed Random Variables\n",
    "\n",
    "The discrete Bayes filter works by multiplying and adding arbitrary probability random variables. The Kalman filter uses Gaussians instead of arbitrary random variables, but the rest of the algorithm remains the same. This means we will need to multiply and add Gaussian random variables (Gaussian random variable is just another way to say normally distributed random variable). \n",
    "\n",
    "A remarkable property of Gaussian random variables is that the sum of two independent Gaussian random variables is also normally distributed! The product is not Gaussian, but proportional to a Gaussian. There we can say that the result of multipying two Gaussian distributions is a Gaussian function (recall function in this context means that the property that the values sum to one is not guaranteed).\n",
    "\n",
    "Wikipedia has a good article on this property, and I also prove it at the end of this chapter. \n",
    "https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables\n",
    "\n",
    "Before we do the math, let's test this visually. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(-1, 3, 0.01)\n",
    "g1 = gaussian(x, mean=0.8, var=.1)\n",
    "g2 = gaussian(x, mean=1.3, var=.2)\n",
    "plt.plot(x, g1, x, g2)\n",
    "\n",
    "g = g1 * g2  # element-wise multiplication\n",
    "g = g / sum(g)  # normalize\n",
    "plt.plot(x, g, ls='-.');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here I created two Gaussians, g1=$\\mathcal N(0.8, 0.1)$ and g2=$\\mathcal N(1.3, 0.2)$ and plotted them. Then I multiplied them together and normalized the result. As you can see the result *looks* like a Gaussian distribution.\n",
    "\n",
    "Gaussians are nonlinear functions. Typically, if you multiply a nonlinear equations you end up with a different type of function. For example, the shape of multiplying two sins is very different from `sin(x)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(0, 4*np.pi, 0.01)\n",
    "plt.plot(np.sin(1.2*x))\n",
    "plt.plot(np.sin(1.2*x) * np.sin(2*x));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But the result of multiplying two Gaussians distributions is a Gaussian function. This is a key reason why Kalman filters are computationally feasible. Said another way, Kalman filters use Gaussians *because* they are computationally nice. \n",
    "\n",
    "The product of two independent Gaussians is given by:\n",
    "\n",
    "$$\\begin{aligned}\\mu &=\\frac{\\sigma_1^2\\mu_2 + \\sigma_2^2\\mu_1}{\\sigma_1^2+\\sigma_2^2}\\\\\n",
    "\\sigma^2 &=\\frac{\\sigma_1^2\\sigma_2^2}{\\sigma_1^2+\\sigma_2^2} \n",
    "\\end{aligned}$$\n",
    "\n",
    "The sum of two Gaussian random variables is given by\n",
    "\n",
    "$$\\begin{gathered}\\mu = \\mu_1 + \\mu_2 \\\\\n",
    "\\sigma^2 = \\sigma^2_1 + \\sigma^2_2\n",
    "\\end{gathered}$$\n",
    "\n",
    "At the end of the chapter I derive these equations. However, understanding the deriviation is not very important."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Putting it all Together\n",
    "\n",
    "Now we are ready to talk about how Gaussians can be used in filtering. In the next chapter we will implement a filter using Gaussins. Here I will explain why we would want to use Gaussians.\n",
    "\n",
    "In the previous chapter we represented probability distributions with an array. We performed the update computation by computing the element-wise product of that distribution with another distribution representing the likelihood of the measurement at each point, like so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def normalize(p):\n",
    "    return p / sum(p)\n",
    "\n",
    "def update(likelihood, prior):\n",
    "    return normalize(likelihood * prior)\n",
    "\n",
    "prior =      normalize(np.array([4, 2, 0, 7, 2, 12, 35, 20, 3, 2]))\n",
    "likelihood = normalize(np.array([3, 4, 1, 4, 2, 38, 20, 18, 1, 16]))\n",
    "posterior = update(likelihood, prior)\n",
    "book_plots.bar_plot(posterior)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words, we have to compute 10 multiplications to get this result. For a real filter with large arrays in multiple dimensions we'd require billions of multiplications, and vast amounts of memory. \n",
    "\n",
    "But this distribution looks like a Gaussian. What if we use a Gaussian instead of an array? I'll compute the mean and variance of the posterior and plot it against the bar chart."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean: 5.88 var: 1.24\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = np.arange(0, 10, .01)\n",
    "\n",
    "def mean_var(p):\n",
    "    x = np.arange(len(p))\n",
    "    mean = np.sum(p * x,dtype=float)\n",
    "    var = np.sum((x - mean)**2 * p)\n",
    "    return mean, var\n",
    "\n",
    "mean, var = mean_var(posterior)\n",
    "book_plots.bar_plot(posterior)\n",
    "plt.plot(xs, gaussian(xs, mean, var, normed=False), c='r');\n",
    "print('mean: %.2f' % mean, 'var: %.2f' % var)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is impressive. We can describe an entire distribution of numbers with only two numbers. Perhaps this example is not persuasive, given there are only 10 numbers in the distribution. But a real problem could have millions of numbers, yet still only require two numbers to describe it.\n",
    "\n",
    "Next, recall that our filter implements the update function with\n",
    "```python\n",
    "def update(likelihood, prior):\n",
    "    return normalize(likelihood * prior)\n",
    "```\n",
    "\n",
    "If the arrays contain a million elements, that is one million multiplications. However, if we replace the arrays with a Gaussian then we would perform that calculation with\n",
    "\n",
    "$$\\begin{aligned}\\mu &=\\frac{\\sigma_1^2\\mu_2 + \\sigma_2^2\\mu_1}{\\sigma_1^2+\\sigma_2^2}\\\\\n",
    "\\sigma^2 &=\\frac{\\sigma_1^2\\sigma_2^2}{\\sigma_1^2+\\sigma_2^2} \n",
    "\\end{aligned}$$\n",
    "\n",
    "which is three multiplications and two divisions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bayes Theorem\n",
    "\n",
    "In the last chapter we developed an algorithm by reasoning about the information we have at each moment, which we expressed as discrete probability distributions. In the process we discovered [*Bayes' Theorem*](https://en.wikipedia.org/wiki/Bayes%27_theorem). Bayes theorem tells us how to compute the probability of an event given prior information. \n",
    "\n",
    "We implemented the `update()` function with this probability calculation:\n",
    "\n",
    "$$ \\mathtt{posterior} = \\frac{\\mathtt{likelihood}\\times \\mathtt{prior}}{\\mathtt{normalization}}$$ \n",
    "\n",
    "It turns out that this is Bayes' theorem. In a second I will develop the mathematics, but in many ways that obscures the simple idea expressed in this equation. We read this as:\n",
    "\n",
    "$$ updated\\,knowledge = \\big\\|likelihood\\,of\\,new\\,knowledge\\times prior\\, knowledge \\big\\|$$\n",
    "\n",
    "where $\\| \\cdot\\|$ expresses normalizing the term.\n",
    "\n",
    "We came to this with simple reasoning about a dog walking down a hallway. Yet, as we will see, the same equation applies to a universe of filtering problems. We will use this equation in every subsequent chapter.\n",
    "\n",
    "\n",
    "To review, the *prior* is the probability of something happening before we include the probability of the measurement (the *likelihood*) and the *posterior* is the probability we compute after incorporating the information from the measurement.\n",
    "\n",
    "Bayes theorem is\n",
    "\n",
    "$$P(A \\mid B) = \\frac{P(B \\mid A)\\, P(A)}{P(B)}$$\n",
    "\n",
    "$P(A \\mid B)$ is called a [*conditional probability*](https://en.wikipedia.org/wiki/Conditional_probability). That is, it represents the probability of $A$ happening *if* $B$ happened. For example, it is more likely to rain today compared to a typical day if it also rained yesterday because rain systems usually last more than one day. We'd write the probability of it raining today given that it rained yesterday as $P$(rain today $\\mid$ rain yesterday).\n",
    "\n",
    "\n",
    "I've glossed over an important point. In our code above we are not working with single probabilities, but an array of probabilities - a *probability distribution*. The equation I just gave for Bayes uses probabilities, not probability distributions. However, it is equally valid with probability distributions. We use a lower case $p$ for probability distributions\n",
    "\n",
    "$$p(A \\mid B) = \\frac{p(B \\mid A)\\, p(A)}{p(B)}$$\n",
    "\n",
    "In the equation above $B$ is the *evidence*, $p(A)$ is the *prior*, $p(B \\mid A)$ is the *likelihood*, and $p(A \\mid B)$ is the *posterior*. By substituting the mathematical terms with the corresponding words you can see that Bayes theorem matches our update equation. Let's rewrite the equation in terms of our problem. We will use $x_i$ for the position at *i*, and $z$ for the measurement. Hence, we want to know $P(x_i \\mid z)$, that is, the probability of the dog being at $x_i$ given the measurement $z$. \n",
    "\n",
    "So, let's plug that into the equation and solve it.\n",
    "\n",
    "$$p(x_i \\mid z) = \\frac{p(z \\mid x_i) p(x_i)}{p(z)}$$\n",
    "\n",
    "That looks ugly, but it is actually quite simple. Let's figure out what each term on the right means. First is $p(z \\mid x_i)$. This is the likelihood, or the probability for the measurement at every cell $x_i$. $p(x_i)$ is the *prior* - our belief before incorporating the measurements. We multiply those together. This is just the unnormalized multiplication in the `update()` function:\n",
    "\n",
    "```python\n",
    "def update(likelihood, prior):\n",
    "    posterior = prior * likelihood   # p(z|x) * p(x)\n",
    "    return normalize(posterior)\n",
    "```\n",
    "\n",
    "The last term to consider is the denominator $p(z)$. This is the probability of getting the measurement $z$ without taking the location into account. It is often called the *evidence*. We compute that by taking the sum of $x$, or `sum(belief)` in the code. That is how we compute the normalization! So, the `update()` function is doing nothing more than computing Bayes' theorem.\n",
    "\n",
    "The literature often gives you these equations in the form of integrals. After all, an integral is just a sum over a continuous function. So, you might see Bayes' theorem written as\n",
    "\n",
    "$$p(A \\mid B) = \\frac{p(B \\mid A)\\, p(A)}{\\int p(B \\mid A_j) p(A_j) \\,\\, \\mathtt{d}A_j}\\cdot$$\n",
    "\n",
    "This denominator is usually impossible to solve analytically; when it can be solved the math is fiendishly difficult. A recent [opinion piece ](http://www.statslife.org.uk/opinion/2405-we-need-to-rethink-how-we-teach-statistics-from-the-ground-up)for the Royal Statistical Society called it a \"dog's breakfast\" [8].  Filtering textbooks that take a Bayesian approach are filled with integral laden equations with no analytic solution. Do not be cowed by these equations, as we trivially handled this integral by normalizing our posterior. We will learn more techniques to handle this in the **Particle Filters** chapter. Until then, recognize that in practice it is just a normalization term over which we can sum. What I'm trying to say is that when you are faced with a page of integrals, just think of them as sums, and relate them back to this chapter, and often the difficulties will fade. Ask yourself \"why are we summing these values\", and \"why am I dividing by this term\". Surprisingly often the answer is readily apparent. Surprisingly often the author neglects to mention this interpretation.\n",
    "\n",
    "It's probable that the strength of Bayes' theorem is not yet fully apparent to you. We want to compute $p(x_i \\mid Z)$. That is, at step i, what is our probable state given a measurement. That's an extraordinarily difficult problem in general. Bayes' Theorem is general. We may want to know the probability that we have cancer given the results of a cancer test, or the probability of rain given various sensor readings. Stated like that the problems seem unsolvable.\n",
    "\n",
    "But Bayes' Theorem lets us compute this by using the inverse  $p(Z\\mid x_i)$, which is often straightforward to compute\n",
    "\n",
    "$$p(x_i \\mid Z) \\propto p(Z\\mid x_i)\\, p(x_i)$$\n",
    "\n",
    "That is, to compute how likely it is to rain given specific sensor readings we only have to compute the likelihood of the sensor readings given that it is raining! That's a ***much*** easier problem! Well, weather prediction is still a difficult problem, but Bayes makes it tractable. \n",
    "\n",
    "Likewise, as you saw in the Discrete Bayes chapter, we computed the likelihood that Simon was in any given part of the hallway by computing how likely a sensor reading is given that Simon is at position `x`. A hard problem becomes easy. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Total Probability Theorem\n",
    "\n",
    "We now know the formal mathematics behind the `update()` function; what about the `predict()` function? `predict()` implements the [*total probability theorem*](https://en.wikipedia.org/wiki/Law_of_total_probability). Let's recall what `predict()` computed. It computed the probability of being at any given position given the probability of all the possible movement events. Let's express that as an equation. The probability of being at any position $i$ at time $t$ can be written as $P(X_i^t)$. We computed that as the sum of the prior at time $t-1$ $P(X_j^{t-1})$ multiplied by the probability of moving from cell $x_j$ to $x_i$. That is\n",
    "\n",
    "$$P(X_i^t) = \\sum_j P(X_j^{t-1})  P(x_i | x_j)$$\n",
    "\n",
    "That equation is called the *total probability theorem*. Quoting from Wikipedia [6] \"It expresses the total probability of an outcome which can be realized via several distinct events\". I could have given you that equation and implemented `predict()`, but your chances of understanding why the equation works would be slim. As a reminder, here is the code that computes this equation\n",
    "\n",
    "```python\n",
    "for i in range(N):\n",
    "    for k in range (kN):\n",
    "        index = (i + (width-k) - offset) % N\n",
    "        result[i] += prob_dist[index] * kernel[k]\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing Probabilities with scipy.stats\n",
    "\n",
    "In this chapter I used code from [FilterPy](https://github.com/rlabbe/filterpy) to compute and plot Gaussians. I did that to give you a chance to look at the code and see how these functions are implemented.  However, Python comes with \"batteries included\" as the saying goes, and it comes with a wide range of statistics functions in the module `scipy.stats`. So let's walk through how to use scipy.stats to compute statistics and probabilities.\n",
    "\n",
    "The `scipy.stats` module contains a number of objects which you can use to compute attributes of various probability distributions. The full documentation for this module is here: http://docs.scipy.org/doc/scipy/reference/stats.html. We will focus on the  norm variable, which implements the normal distribution. Let's look at some code that uses `scipy.stats.norm` to compute a Gaussian, and compare its value to the value returned by the `gaussian()` function from FilterPy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.13114657203397997\n",
      "0.13114657203397995\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "import filterpy.stats\n",
    "print(norm(2, 3).pdf(1.5))\n",
    "print(filterpy.stats.gaussian(x=1.5, mean=2, var=3*3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The call `norm(2, 3)` creates what scipy calls a 'frozen' distribution - it creates and returns an object with a mean of 2 and a standard deviation of 3. You can then use this object multiple times to get the probability density of various values, like so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pdf of 1.5 is       0.1311\n",
      "pdf of 2.5 is also  0.1311\n",
      "pdf of 2 is         0.1330\n"
     ]
    }
   ],
   "source": [
    "n23 = norm(2, 3)\n",
    "print('pdf of 1.5 is       %.4f' % n23.pdf(1.5))\n",
    "print('pdf of 2.5 is also  %.4f' % n23.pdf(2.5))\n",
    "print('pdf of 2 is         %.4f' % n23.pdf(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The documentation for  [scipy.stats.norm](http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.normfor) [2] lists many other functions. For example, we can generate $n$ samples from the distribution with the `rvs()` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 2.269  2.485 -0.469  1.991  2.468  1.667  1.231\n",
      "  4.577  2.56   0.224 -3.851  2.604  3.583 -1.623\n",
      "  2.348]\n"
     ]
    }
   ],
   "source": [
    "np.set_printoptions(precision=3, linewidth=50)\n",
    "print(n23.rvs(size=15))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can get the [*cumulative distribution function (CDF)*](https://en.wikipedia.org/wiki/Cumulative_distribution_function), which is the probability that a randomly drawn value from the distribution is less than or equal to $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5\n"
     ]
    }
   ],
   "source": [
    "# probability that a random value is less than the mean 2\n",
    "print(n23.cdf(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can get various properties of the distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "variance is 9.0\n",
      "standard deviation is 3.0\n",
      "mean is 2.0\n"
     ]
    }
   ],
   "source": [
    "print('variance is', n23.var())\n",
    "print('standard deviation is', n23.std())\n",
    "print('mean is', n23.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limitations of Using Gaussians to Model the World\n",
    "\n",
    "Earlier I mentioned the *central limit theorem*, which states that under certain conditions the arithmetic sum of any independent random variable will be normally distributed, regardless of how the random variables are distributed. This is important to us because nature is full of distributions which are not normal, but when we apply the central limit theorem over large populations we end up with normal distributions. \n",
    "\n",
    "However, a key part of the proof is “under certain conditions”. These conditions often do not hold for the physical world. For example, a kitchen scale cannot read below zero, but if we represent the measurement error as a Gaussian the left side of the curve extends to negative infinity, implying a very small chance of giving a negative reading. \n",
    "\n",
    "This is a broad topic which I will not treat exhaustively. \n",
    "\n",
    "Let's consider a trivial example. We think of things like test scores as being normally distributed. If you have ever had a professor “grade on a curve” you have been subject to this assumption. But of course test scores cannot follow a normal distribution. This is because the distribution assigns a nonzero probability distribution for *any* value, no matter how far from the mean. So, for example, say your mean is 90 and the standard deviation is 13. The normal distribution assumes that there is a large chance of somebody getting a 90, and a small chance of somebody getting a 40. However, it also implies that there is a tiny chance of somebody getting a grade of -10, or 150. It assigns an extremely small chance of getting a score of $-10^{300}$ or $10^{32986}$. The tails of a Gaussian distribution are infinitely long.\n",
    "\n",
    "But for a test we know this is not true. Ignoring extra credit, you cannot get less than 0, or more than 100. Let's plot this range of values using a normal distribution to see how poorly this represents real test scores distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = np.arange(10, 100, 0.05)\n",
    "ys = [gaussian(x, 90, 30) for x in xs]\n",
    "plt.plot(xs, ys, label='var=0.2')\n",
    "plt.xlim(0, 120)\n",
    "plt.ylim(-0.02, 0.09);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The area under the curve cannot equal 1, so it is not a probability distribution. What actually happens is that more students than predicted by a normal distribution get scores nearer the upper end of the range (for example), and that tail becomes “fat”. Also, the test is probably not able to perfectly distinguish minute differences in skill in the students, so the distribution to the left of the mean is also probably a bit bunched up in places. \n",
    "\n",
    "Sensors measure the world. The errors in a sensor's measurements are rarely truly Gaussian. It is far too early to be talking about the difficulties that this presents to the Kalman filter designer. It is worth keeping in the back of your mind the fact that the Kalman filter math is based on an idealized model of the world.  For now I will present a bit of code that I will be using later in the book to form distributions to simulate various processes and sensors. This distribution is called the [*Student's $t$-distribution*](https://en.wikipedia.org/wiki/Student%27s_t-distribution). \n",
    "\n",
    "Let's say I want to model a sensor that has some white noise in the output. For simplicity, let's say the signal is a constant 10, and the standard deviation of the noise is 2. We can use the function `numpy.random.randn()` to get a random number with a mean of 0 and a standard deviation of 1. I can simulate this with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.random import randn\n",
    "def sense():\n",
    "    return 10 + randn()*2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot that signal and see what it looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "zs = [sense() for i in range(5000)]\n",
    "plt.plot(zs, lw=1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That looks like what I would expect. The signal is centered around 10. A standard deviation of 2 means that 68% of the measurements will be within $\\pm$ 2 of 10, and 99% will be within $\\pm$ 6 of 10, and that looks like what is happening. \n",
    "\n",
    "Now let's look at distribution generated with the Student's $t$-distribution. I will not go into the math, but just give you the source code for it and then plot a distribution using it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "import math\n",
    "\n",
    "def rand_student_t(df, mu=0, std=1):\n",
    "    \"\"\"return random number distributed by Student's t \n",
    "    distribution with `df` degrees of freedom with the \n",
    "    specified mean and standard deviation.\n",
    "    \"\"\"\n",
    "    x = random.gauss(0, std)\n",
    "    y = 2.0*random.gammavariate(0.5*df, 2.0)\n",
    "    return x / (math.sqrt(y / df)) + mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def sense_t():\n",
    "    return 10 + rand_student_t(7)*2\n",
    "\n",
    "zs = [sense_t() for i in range(5000)]\n",
    "plt.plot(zs, lw=1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see from the plot that while the output is similar to the normal distribution there are outliers that go far more than 3 standard deviations from the mean (7 to 13). \n",
    "\n",
    "It is unlikely that the Student's $t$-distribution is an accurate model of how your sensor (say, a GPS or Doppler) performs, and this is not a book on how to model physical systems. However, it does produce reasonable data to test your filter's performance when presented with real world noise. We will be using distributions like these throughout the rest of the book in our simulations and tests. \n",
    "\n",
    "This is not an idle concern. The Kalman filter equations assume the noise is normally distributed, and perform sub-optimally if this is not true. Designers for mission critical filters, such as the filters on spacecraft, need to master a lot of theory and empirical knowledge about the performance of the sensors on their spacecraft. For example, a presentation I saw on a NASA mission stated that while theory states that they should use 3 standard deviations to distinguish noise from valid measurements in practice they had to use 5 to 6 standard deviations. This was something they determined by experiments.\n",
    "\n",
    "The code for rand_student_t is included in `filterpy.stats`. You may use it with\n",
    "\n",
    "```python\n",
    "from filterpy.stats import rand_student_t\n",
    "```\n",
    "\n",
    "While I'll not cover it here, statistics has defined ways of describing the shape of a probability distribution by how it varies from an exponential distribution. The normal distribution is shaped symmetrically around the mean - like a bell curve. However, a probability distribution can be asymmetrical around the mean. The measure of this is called [*skew*](https://en.wikipedia.org/wiki/Skewness). The tails can be shortened, fatter, thinner, or otherwise shaped differently from an exponential distribution. The measure of this is called [*kurtosis*](https://en.wikipedia.org/wiki/Kurtosis). the `scipy.stats` module contains the function `describe` which computes these statistics, among others."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DescribeResult(nobs=5000, minmax=(-0.19143741858944452, 20.804030812682086), mean=9.996901102331591, variance=2.7388116003856884, skewness=0.015276194207011526, kurtosis=1.622567142788295)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import scipy\n",
    "scipy.stats.describe(zs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's examine two normal populations, one small, one large:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "DescribeResult(nobs=10, minmax=(-1.2037202953168122, 0.894366248899801), mean=-0.18007426896541692, variance=0.43204122396032946, skewness=-0.08295117992362264, kurtosis=-0.9197042651911951)\n",
      "\n",
      "DescribeResult(nobs=300000, minmax=(-4.75206637131407, 4.178833173976851), mean=-0.00047436810878343096, variance=0.9973760542573228, skewness=-0.0038868831924059035, kurtosis=-0.005797621180390955)\n"
     ]
    }
   ],
   "source": [
    "print(scipy.stats.describe(np.random.randn(10)))\n",
    "print()\n",
    "print(scipy.stats.describe(np.random.randn(300000)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The small sample has very non-zero skew and kurtosis because the small number of samples is not well distributed around the mean of 0. You can see this also by comparing the computed mean and variance with the theoretical mean of 0 and variance 1. In comparison the large sample's mean and variance are very close to the theoretical values, and both the skew and kurtosis are near zero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Product of Gaussians (Optional)\n",
    "\n",
    "It is not important to read this section. Here I derive the equations for the product of two Gaussians.\n",
    "\n",
    "You can find this result by multiplying the equation for two Gaussians together and combining terms. The algebra gets messy. I will derive it using Bayes theorem. We can state the problem as: let the prior be $N(\\bar\\mu, \\bar\\sigma^2)$, and measurement be $z \\propto N(z, \\sigma_z^2)$. What is the posterior  x given the measurement z?\n",
    "\n",
    "Write the posterior as $p(x \\mid z)$. Now we can use Bayes Theorem to state\n",
    "\n",
    "$$p(x \\mid z) = \\frac{p(z \\mid x)p(x)}{p(z)}$$\n",
    "\n",
    "$p(z)$ is a normalizing constant, so we can create a proportinality\n",
    "\n",
    "$$p(x \\mid z) \\propto p(z|x)p(x)$$\n",
    "\n",
    "Now we subtitute in the equations for the Gaussians, which are\n",
    "\n",
    "$$p(z \\mid x) = \\frac{1}{\\sqrt{2\\pi\\sigma_z^2}}\\exp \\Big[-\\frac{(z-x)^2}{2\\sigma_z^2}\\Big]$$\n",
    "\n",
    "$$p(x) = \\frac{1}{\\sqrt{2\\pi\\bar\\sigma^2}}\\exp \\Big[-\\frac{(x-\\bar\\mu)^2}{2\\bar\\sigma^2}\\Big]$$\n",
    "\n",
    "We can drop the leading terms, as they are constants, giving us\n",
    "\n",
    "$$\\begin{aligned}\n",
    "p(x \\mid z) &\\propto \\exp \\Big[-\\frac{(z-x)^2}{2\\sigma_z^2}\\Big]\\exp \\Big[-\\frac{(x-\\bar\\mu)^2}{2\\bar\\sigma^2}\\Big]\\\\\n",
    "&\\propto \\exp \\Big[-\\frac{(z-x)^2}{2\\sigma_z^2}-\\frac{(x-\\bar\\mu)^2}{2\\bar\\sigma^2}\\Big] \\\\\n",
    "&\\propto \\exp \\Big[-\\frac{1}{2\\sigma_z^2\\bar\\sigma^2}[\\bar\\sigma^2(z-x)^2+\\sigma_z^2(x-\\bar\\mu)^2]\\Big]\n",
    "\\end{aligned}$$\n",
    "\n",
    "Now we multiply out the squared terms and group in terms of the posterior $x$.\n",
    "\n",
    "$$\\begin{aligned}\n",
    "p(x \\mid z) &\\propto \\exp \\Big[-\\frac{1}{2\\sigma_z^2\\bar\\sigma^2}[\\bar\\sigma^2(z^2 -2xz + x^2) + \\sigma_z^2(x^2 - 2x\\bar\\mu+\\bar\\mu^2)]\\Big ] \\\\\n",
    "&\\propto \\exp \\Big[-\\frac{1}{2\\sigma_z^2\\bar\\sigma^2}[x^2(\\bar\\sigma^2+\\sigma_z^2)-2x(\\sigma_z^2\\bar\\mu + \\bar\\sigma^2z) + (\\bar\\sigma^2z^2+\\sigma_z^2\\bar\\mu^2)]\\Big ]\n",
    "\\end{aligned}$$\n",
    "\n",
    "The last parentheses do not contain the posterior $x$, so it can be treated as a constant and discarded.\n",
    "\n",
    "$$p(x \\mid z) \\propto \\exp \\Big[-\\frac{1}{2}\\frac{x^2(\\bar\\sigma^2+\\sigma_z^2)-2x(\\sigma_z^2\\bar\\mu + \\bar\\sigma^2z)}{\\sigma_z^2\\bar\\sigma^2}\\Big ]\n",
    "$$\n",
    "\n",
    "Divide numerator and denominator by $\\bar\\sigma^2+\\sigma_z^2$ to get\n",
    "\n",
    "$$p(x \\mid z) \\propto \\exp \\Big[-\\frac{1}{2}\\frac{x^2-2x(\\frac{\\sigma_z^2\\bar\\mu + \\bar\\sigma^2z}{\\bar\\sigma^2+\\sigma_z^2})}{\\frac{\\sigma_z^2\\bar\\sigma^2}{\\bar\\sigma^2+\\sigma_z^2}}\\Big ]\n",
    "$$\n",
    "\n",
    "Proportionality allows us create or delete constants at will, so we can factor this into\n",
    "\n",
    "$$p(x \\mid z) \\propto \\exp \\Big[-\\frac{1}{2}\\frac{(x-\\frac{\\sigma_z^2\\bar\\mu + \\bar\\sigma^2z}{\\bar\\sigma^2+\\sigma_z^2})^2}{\\frac{\\sigma_z^2\\bar\\sigma^2}{\\bar\\sigma^2+\\sigma_z^2}}\\Big ]\n",
    "$$\n",
    "\n",
    "A Gaussian is\n",
    "\n",
    "$$N(\\mu,\\, \\sigma^2) \\propto \\exp\\Big [-\\frac{1}{2}\\frac{(x - \\mu)^2}{\\sigma^2}\\Big ]$$\n",
    "\n",
    "So we can see that $p(x \\mid z)$ has a mean of\n",
    "\n",
    "$$\\mu_\\mathtt{posterior} = \\frac{\\sigma_z^2\\bar\\mu + \\bar\\sigma^2z}{\\bar\\sigma^2+\\sigma_z^2}$$\n",
    "\n",
    "and a variance of\n",
    "$$\n",
    "\\sigma_\\mathtt{posterior} = \\frac{\\sigma_z^2\\bar\\sigma^2}{\\bar\\sigma^2+\\sigma_z^2}\n",
    "$$\n",
    "\n",
    "I've dropped the constants, and so the result is not a normal, but proportional to one. Bayes theorem normalizes with the $p(z)$ divisor, ensuring that the result is normal. We normalize in the update step of our filters, ensuring the filter estimate is Gaussian.\n",
    "\n",
    "$$\\mathcal N_1 = \\| \\mathcal N_2\\cdot \\mathcal N_3\\|$$"
   ]
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    "## Sum of Independent Gaussian Random Variables  (Optional)\n",
    "\n",
    "Likewise, this section is not important to read. Here I derive the equations for the sum of two independent Gaussian random variables. \n",
    "\n",
    "The sum is given by\n",
    "\n",
    "$$\\begin{gathered}\\mu = \\mu_1 + \\mu_2 \\\\\n",
    "\\sigma^2 = \\sigma^2_1 + \\sigma^2_2\n",
    "\\end{gathered}$$\n",
    "\n",
    "There are several proofs for this. I will use convolution since we used convolution in the previous chapter for the histograms of probabilities. \n",
    "\n",
    "To find the density function of the sum of two Gaussian random variables we sum the density functions of each. They are nonlinear, continuous functions, so we need to compute the sum with an integral. If the random variables $p$ and $z$ (e.g. prior and measurement) are independent we can compute this with\n",
    "\n",
    "$p(x) = \\int\\limits_{-\\infty}^\\infty f_p(x-z)f_z(z)\\, dx$\n",
    "\n",
    "This is the equation for a convolution. Now we just do some math:\n",
    "\n",
    "\n",
    "$p(x) = \\int\\limits_{-\\infty}^\\infty f_2(x-x_1)f_1(x_1)\\, dx$\n",
    "\n",
    "$=  \\int\\limits_{-\\infty}^\\infty \n",
    "\\frac{1}{\\sqrt{2\\pi}\\sigma_z}\\exp\\left[-\\frac{(x - z - \\mu_z)^2}{2\\sigma^2_z}\\right]\n",
    "\\frac{1}{\\sqrt{2\\pi}\\sigma_p}\\exp\\left[-\\frac{(x - \\mu_p)^2}{2\\sigma^2_p}\\right] \\, dx$\n",
    "\n",
    "$=  \\int\\limits_{-\\infty}^\\infty\n",
    "\\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma_p^2 + \\sigma_z^2}} \\exp\\left[ -\\frac{(x - (\\mu_p + \\mu_z)))^2}{2(\\sigma_z^2+\\sigma_p^2)}\\right]\n",
    "\\frac{1}{\\sqrt{2\\pi}\\frac{\\sigma_p\\sigma_z}{\\sqrt{\\sigma_p^2 + \\sigma_z^2}}} \\exp\\left[ -\\frac{(x - \\frac{\\sigma_p^2(x-\\mu_z) + \\sigma_z^2\\mu_p}{}))^2}{2\\left(\\frac{\\sigma_p\\sigma_x}{\\sqrt{\\sigma_z^2+\\sigma_p^2}}\\right)^2}\\right] \\, dx$\n",
    "\n",
    "$= \\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma_p^2 + \\sigma_z^2}} \\exp\\left[ -\\frac{(x - (\\mu_p + \\mu_z)))^2}{2(\\sigma_z^2+\\sigma_p^2)}\\right] \\int\\limits_{-\\infty}^\\infty\n",
    "\\frac{1}{\\sqrt{2\\pi}\\frac{\\sigma_p\\sigma_z}{\\sqrt{\\sigma_p^2 + \\sigma_z^2}}} \\exp\\left[ -\\frac{(x - \\frac{\\sigma_p^2(x-\\mu_z) + \\sigma_z^2\\mu_p}{}))^2}{2\\left(\\frac{\\sigma_p\\sigma_x}{\\sqrt{\\sigma_z^2+\\sigma_p^2}}\\right)^2}\\right] \\, dx$\n",
    "\n",
    "The expression inside the integral is a normal distribution. The sum of a normal distribution is one, hence the integral is one. This gives us\n",
    "\n",
    "$$p(x) = \\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma_p^2 + \\sigma_z^2}} \\exp\\left[ -\\frac{(x - (\\mu_p + \\mu_z)))^2}{2(\\sigma_z^2+\\sigma_p^2)}\\right]$$\n",
    "\n",
    "This is in the form of a normal, where\n",
    "\n",
    "$$\\begin{gathered}\\mu_x = \\mu_p + \\mu_z \\\\\n",
    "\\sigma_x^2 = \\sigma_z^2+\\sigma_p^2\\, \\square\\end{gathered}$$"
   ]
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   "source": [
    "## Summary and Key Points\n",
    "\n",
    "This chapter is a poor introduction to statistics in general. I've only covered the concepts that  needed to use Gaussians in the remainder of the book, no more. What I've covered will not get you very far if you intend to read the Kalman filter literature. If this is a new topic to you I suggest reading a statistics textbook. I've always liked the Schaum series for self study, and Alan Downey's *Think Stats* [5] is also very good and freely available online. \n",
    "\n",
    "The following points **must** be understood by you before we continue:\n",
    "\n",
    "* Normals express a continuous probability distribution\n",
    "* They are completely described by two parameters: the mean ($\\mu$) and variance ($\\sigma^2$)\n",
    "* $\\mu$ is the average of all possible values\n",
    "* The variance $\\sigma^2$ represents how much our measurements vary from the mean\n",
    "* The standard deviation ($\\sigma$) is the square root of the variance ($\\sigma^2$)\n",
    "* Many things in nature approximate a normal distribution, but the math is not perfect.\n",
    "* In filtering problems computing $p(x\\mid z)$ is nearly impossible, but computing $p(z\\mid x)$ is straightforward. Bayes' lets us compute the former from the latter. \n",
    "\n",
    "The next several chapters will be using Gaussians with Bayes' theorem to help perform filtering. As noted in the last section, sometimes Gaussians do not describe the world very well. Latter parts of the book are dedicated to filters which work even when the noise or system's behavior is very non-Gaussian. "
   ]
  },
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   "metadata": {},
   "source": [
    "## References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[1] https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Supporting_Notebooks/Computing_and_plotting_PDFs.ipynb\n",
    "\n",
    "[2] http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html\n",
    "\n",
    "[3] http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html\n",
    "\n",
    "[4] Huber, Peter J. *Robust Statistical Procedures*, Second Edition. Society for Industrial and Applied Mathematics, 1996.\n",
    "\n",
    "[5] Downey, Alan. *Think Stats*, Second Edition. O'Reilly Media.\n",
    "\n",
    "https://github.com/AllenDowney/ThinkStats2\n",
    "\n",
    "http://greenteapress.com/thinkstats/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Useful Wikipedia Links\n",
    "\n",
    "https://en.wikipedia.org/wiki/Probability_distribution\n",
    "\n",
    "https://en.wikipedia.org/wiki/Random_variable\n",
    "\n",
    "https://en.wikipedia.org/wiki/Sample_space\n",
    "\n",
    "https://en.wikipedia.org/wiki/Central_tendency\n",
    "\n",
    "https://en.wikipedia.org/wiki/Expected_value\n",
    "\n",
    "https://en.wikipedia.org/wiki/Standard_deviation\n",
    "\n",
    "https://en.wikipedia.org/wiki/Variance\n",
    "\n",
    "https://en.wikipedia.org/wiki/Probability_density_function\n",
    "\n",
    "https://en.wikipedia.org/wiki/Central_limit_theorem\n",
    "\n",
    "https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule\n",
    "\n",
    "https://en.wikipedia.org/wiki/Cumulative_distribution_function\n",
    "\n",
    "https://en.wikipedia.org/wiki/Skewness\n",
    "\n",
    "https://en.wikipedia.org/wiki/Kurtosis"
   ]
  }
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