statistics/10_days/18_day5normaldistributioni.ipynb

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   "source": [
    "Solution to: [Day 5: Normal Distribution I](https://www.hackerrank.com/challenges/s10-normal-distribution-1/tutorial)"
   ]
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    "<h1 id=\"tocheading\">Table of Contents</h1>\n",
    "<div id=\"toc\"></div>\n",
    "\n",
    "- Table of Contents\n",
    "- Notes\n",
    "    - Normal Distribution\n",
    "    - Standard Normal Distribution\n",
    "    - Cumulative Probability\n",
    "- Solution\n",
    "    - Imports\n",
    "    - Constants\n",
    "    - Input\n",
    "    - Cumulative Probability function\n",
    "    - Main"
   ]
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    "# Notes\n",
    "\n",
    "## Normal Distribution\n",
    "The probability density of normal distribution is:\n",
    "\\begin{equation}\n",
    "\\large\n",
    "\\mathcal{N} = \\frac\n",
    "{1}\n",
    "{\\sigma * 2 \\pi}\n",
    "e^\n",
    "{\\frac\n",
    "{(x - \\mu)^{2}}\n",
    "{2\\sigma^{2}}\n",
    "}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "Here,\n",
    "- μ is the mean (or expectation) of the distribution.\n",
    "    - It can also be the median & mode.\n",
    "- $\\sigma^{2}$ is the variance.\n",
    "- $\\sigma$ is the standard deviation."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Standard Normal Distribution\n",
    "If μ = 0 and σ = 1, then the normal distribution is known as standard normal distribution:\n",
    "\n",
    "\\begin{equation}\n",
    "\\large\n",
    "\\phi(x) = \n",
    "\\frac\n",
    "{e^{-\\frac{x^{2}}{2}}}\n",
    "{\\pi^{\\frac{1}{2}}}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "Every normal distribution can be represented as standard normal distribution:\n",
    "\\begin{equation}\n",
    "\\large\n",
    "\\mathcal{N} (\\mu\\sigma^{2}) = \n",
    "\\frac{1}{\\sigma}\n",
    "\\Phi\n",
    "\\frac\n",
    "{x - \\mu}\n",
    "{\\sigma}\n",
    "\\end{equation}\n"
   ]
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   "metadata": {},
   "source": [
    "## Cumulative Probability\n",
    "Consider a real-valued random variable, X. The **cumulative distribution function** of X (or just the distribution function of X) evaluated at x is the probability that X will take a value less than or equal to x:\n",
    "\n",
    "\\begin{equation}\n",
    "\\large\n",
    "F_{x}(x) = P(X \\leq x)\n",
    "\\end{equation}\n",
    "\t\n",
    "    \n",
    "Also,\n",
    "\\begin{equation}\n",
    "\\large\n",
    "P(A \\leq X \\leq B) = P(A < B) = F_{x}(B) - F_{x}(A)\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "The **cumulative distribution function** for a function with normal distribution is:\n",
    "\n",
    "\\begin{equation}\n",
    "\\large\n",
    "\\Phi(x) = \n",
    "\\frac{1}{2}(1 + erf(\n",
    "\\frac\n",
    "{x - mu}\n",
    "{\\sigma * 2^{\\frac{1}{2}}}\n",
    "))\n",
    "\\end{equation}\n",
    "\n",
    "Where erf is the **error function**:\n",
    "\n",
    "\\begin{equation}\n",
    "\\large\n",
    "(z) = \\frac\n",
    "{2}{\\pi^{\\frac{1}{2}}}\n",
    "\\sum_{0}^{z} e^{-x^{2}} * dx\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "from typing import Tuple"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_input() -> Tuple[int, int, float, ]:\n",
    "\t\"\"\"Returns input for Day 5: Normal Distribution I\n",
    "\n",
    "\tReturns:\n",
    "\t\tTuple[int, int, float, int]: Represents: mean, sd, less than value, and interval\n",
    "\t\"\"\"\n",
    "\treturn (20, 2, 19.5, 20, 22)\n",
    "\n",
    "\tmean, sd = (int(x) for x in input().split())\n",
    "\tless_than = float(input())\n",
    "\tinterval_1, interval_2 = (int(x) for x in input().split())\n",
    "\treturn mean, sd, less_than, interval_1, interval_2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cumulative Probability function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def custom_erf(z: int) -> float:\n",
    "\t\"\"\"Returns error from error function for normal distribution\n",
    "\n",
    "\tArgs:\n",
    "\t\tz (int): error function value\n",
    "\n",
    "\tReturns:\n",
    "\t\tfloat: error amount\n",
    "\t\n",
    "\tNOTE:\n",
    "\t\t- math module has an inbuilt erf function\n",
    "\t\t- dt represents an infinitely small value from a Maclaurin series\n",
    "\t\t- formula taken from: https://www.johndcook.com/blog/python_erf/\n",
    "\t\"\"\"\n",
    "\t## Vars\n",
    "\tconstants = (\n",
    "\t\t1.061405429,\n",
    "\t\t-1.453152027,\n",
    "\t\t1.421413741,\n",
    "\t\t-0.284496736,\n",
    "\t\t0.254829592,\n",
    "\t)\n",
    "\tp = 0.3275911\n",
    "\tsign = 1 if z >= 0 else -1\n",
    "\tz = abs(z)\n",
    "\tt = 1 / ( 1 + p * z)\n",
    "\n",
    "\t## Calc\n",
    "\ty = 0\n",
    "\tfor c in constants: y = (y + c) * t\n",
    "\ty = 1 - (y * math.e ** (-z * z))\n",
    "\n",
    "\treturn y * sign"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calc_cum_distr(x: int, mean: float,\tsd: float) -> float:\n",
    "\t\"\"\"returns cumulative probability for val in normal distribution.\n",
    "\n",
    "\tArgs:\n",
    "\t\tx (int): value to find cum probability for\n",
    "\n",
    "\tReturns:\n",
    "\t\tfloat: cumulative probability of x\n",
    "\t\"\"\"\n",
    "\tz = (x - mean) / (sd * 2 ** (1/2))\n",
    "\treturn 1/2 * (1 + custom_erf(z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def format_scale(num: float) -> float:\n",
    "\t\"\"\"Formats number to scale\n",
    "\n",
    "\tArgs:\n",
    "\t\tnum (float): number\n",
    "\n",
    "\tReturns:\n",
    "\t\tfloat: formatted number\n",
    "\t\"\"\"\n",
    "\treturn f'{num :.3f}'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Main"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "def main():\n",
    "\t\"\"\"Runs script\n",
    "\t\"\"\"\n",
    "\t## Input\n",
    "\tmean, sd, less_than, interval_1, interval_2 = get_input()\n",
    "\n",
    "\t## Cum prob for less than\n",
    "\tcum_prob_less_than = calc_cum_distr(less_than, mean, sd)\n",
    "\tprint( format_scale(cum_prob_less_than))\n",
    "\n",
    "\t## Interval cum prob\n",
    "\tint1_cum_prob = calc_cum_distr(interval_1, mean, sd)\n",
    "\tint2_cum_prob = calc_cum_distr(interval_2, mean, sd)\n",
    "\tinterval_cum_prob = int2_cum_prob - int1_cum_prob\n",
    "\tprint( format_scale(interval_cum_prob))"
   ]
  },
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      "0.401\n",
      "0.341\n"
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   "source": [
    "if __name__ == \"__main__\":\n",
    "\tmain()"
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  }
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