8. Distributions, simulations and histograms.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook shows how to use the Distributions package in order to\n",
    "- use common probability distributions, thier pdfs, cdfs, means etc.\n",
    "- simulate random draws from a distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In case you have not installed it yet, add the Distribuions package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General`\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m git-repo `https://github.com/JuliaRegistries/General.git`\n",
      "\u001b[?25l\u001b[2K\u001b[?25h\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m Installed\u001b[22m\u001b[39m PDMats ──────── v0.9.10\n",
      "\u001b[32m\u001b[1m Installed\u001b[22m\u001b[39m Distributions ─ v0.21.3\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.2/Project.toml`\n",
      " \u001b[90m [31c24e10]\u001b[39m\u001b[92m + Distributions v0.21.3\u001b[39m\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.2/Manifest.toml`\n",
      " \u001b[90m [7d9fca2a]\u001b[39m\u001b[92m + Arpack v0.3.1\u001b[39m\n",
      " \u001b[90m [31c24e10]\u001b[39m\u001b[92m + Distributions v0.21.3\u001b[39m\n",
      " \u001b[90m [90014a1f]\u001b[39m\u001b[92m + PDMats v0.9.10\u001b[39m\n",
      " \u001b[90m [79098fc4]\u001b[39m\u001b[92m + Rmath v0.5.0\u001b[39m\n",
      " \u001b[90m [276daf66]\u001b[39m\u001b[95m ↓ SpecialFunctions v0.8.0 ⇒ v0.7.2\u001b[39m\n",
      " \u001b[90m [4c63d2b9]\u001b[39m\u001b[92m + StatsFuns v0.8.0\u001b[39m\n",
      " \u001b[90m [4607b0f0]\u001b[39m\u001b[92m + SuiteSparse \u001b[39m\n"
     ]
    }
   ],
   "source": [
    "using Pkg\n",
    "Pkg.add(\"Distributions\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we define a distribution, for example, a normal distribution with mean zero and standard deviation 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling Distributions [31c24e10-a181-5473-b8eb-7969acd0382f]\n",
      "└ @ Base loading.jl:1242\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Normal{Float64}(μ=0.0, σ=2.0)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Distributions\n",
    "\n",
    "d = Normal(0,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d is now the normal distribution with mean 2 and standard deviation 2. We can now evaluate, for example, the probability density function of d at some value. Say, we want to know the value of the density at 1.5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.15056871607740221"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pdf(d,1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or the value of the cumulative density function at 1.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8023374568773076"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cdf(d,1.7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#variance of d\n",
    "var(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now suppose you want to draw a random number from the distribution d:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.4371755671505708"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#one random draw from distribution d\n",
    "rand(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or maybe draw 20 times randomly from distribution d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20-element Array{Float64,1}:\n",
       " -1.2337420683416336  \n",
       "  0.012153180902000904\n",
       "  1.4009471072407789  \n",
       " -0.5478666396196415  \n",
       " -1.0519995769444797  \n",
       " -0.6666957960190047  \n",
       "  0.3506775004604972  \n",
       "  0.06317490186181393 \n",
       " -0.29227367331860193 \n",
       " -0.1614865542276003  \n",
       " -0.8673523553307126  \n",
       " -1.5140466843856215  \n",
       "  1.3658428524764445  \n",
       " -0.8572557857852057  \n",
       "  3.395158769438163   \n",
       " -1.7761493523532892  \n",
       "  2.340687068517632   \n",
       "  0.6831994587508208  \n",
       " -3.8027532790080762  \n",
       " -6.366149875226133   "
      ]
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     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "rand(d,20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Histograms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use Plots to plot histograms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "\n",
    "histogram(rand(d,1000))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks already somewhat like normal. If we increase the number of draws, the resulting  sample distribution looks more like the theoretical one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "histogram(rand(d,100000))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Other distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Above we used the normal distributions. Of course, Distributions provides other distributions as well. Below some examples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "du: 3.0 0.5 1.0 0.3333333333333333\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Exponential{Float64}(θ=1.0)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#uniform distribution between 2 and 4\n",
    "du = Uniform(2,4)\n",
    "#print some examples of how to use du\n",
    "println(\"du: \",mean(du),\" \",pdf(du,2.5),\" \",cdf(du,4.0),\" \",var(du))\n",
    "\n",
    "#chi squared distribution with 10 degress of freedom\n",
    "dchi = Chisq(10)\n",
    "\n",
    "#t-distribution with 3 degrees of freedom\n",
    "dt = TDist(3.0)\n",
    "\n",
    "#exponential with scale parameter 1.0\n",
    "de = Exponential(1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There also exist discrete distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6 0.24 0.4\n",
      "6.0 2.4000000000000004 0.11147673600000003\n"
     ]
    }
   ],
   "source": [
    "#Bernoulli with success rate 0.6, i.e. random variable that is 1 with prob 0.6 and 0 with prob 0.4\n",
    "db = Bernoulli(0.6)\n",
    "println(mean(db),\" \",var(db),\" \",pdf(db,0))\n",
    "\n",
    "#Binomial with success rate 0.6 and 10 repetitions\n",
    "dbin = Binomial(10,0.6)\n",
    "println(mean(dbin),\" \",var(dbin),\" \",pdf(dbin,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Application: Intertemporal consumption choice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose a consumer faces the following intertemporal consumption problem. He has a wealth of 100 today and has to decide how much he consumes today and how much he consumes tomorrow (the world ends after tomorrow, i.e. there is only consumption in period 1 or 2 and no other choice). The consumer derives utility $\\sqrt{c}$ by consuming $c$ in a given period and discounts the future with discount factor $0.9$. The interesting bit is that tomorrow he will get a random income $m$ which we assume to be distributed according to $\\chi^2$ distribution with 25 degres of freedom.\n",
    "\n",
    "The consumer's problem is therefore\n",
    "$$\\max_c \\mathbb{E}_m\\left[\\sqrt{c}+0.9*\\sqrt{100-c+m}\\right]$$\n",
    "Of course, one can solve this analytically but here we will go another route. We will solve this problem using Optim and we will evaluate the expectation using integration with quadgk."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Results of Optimization Algorithm\n",
       " * Algorithm: Brent's Method\n",
       " * Search Interval: [0.000000, 100.000000]\n",
       " * Minimizer: 6.870476e+01\n",
       " * Minimum: -1.502850e+01\n",
       " * Iterations: 12\n",
       " * Convergence: max(|x - x_upper|, |x - x_lower|) <= 2*(1.5e-08*|x|+2.2e-16): true\n",
       " * Objective Function Calls: 13"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Optim\n",
    "using QuadGK\n",
    "\n",
    "#define distribution of m\n",
    "md = Chisq(25)\n",
    "\n",
    "function objective(c)\n",
    "    integrand(m) = (sqrt(c)+0.9*sqrt(100-c+m))*pdf(md,m)\n",
    "    return -quadgk(integrand,0,Inf)[1] #\"-\" because Optim searches for a minimizer\n",
    "end\n",
    "\n",
    "optimize(objective,0.0,100.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the result is to consume approximately 68.7 units today and save the rest."
   ]
  }
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