01_Proba_TP.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# TP 01. Probability (NDA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### enseignant: Anastasios Giovanidis 2022 - 2023\n",
    "#### date: 20 Septembre 2022"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### student name or binome:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need to import the following libraries:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The main purpose is to get familiar with:\n",
    "\n",
    "- basic distributions in Python\n",
    "- the law of large numbers (LLN), \n",
    "- the central limit theorem (CLT)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1 (probability functions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Continuous random variables**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(A.1) Generate $N=1000$ samples using the following distributions:\n",
    "\n",
    "- Normal, with $\\mu=2$, $\\sigma=4$\n",
    "- Normal, with $\\mu=2$, $\\sigma^2=4$\n",
    "- Uniform, with $[a,b]=[2,5]$\n",
    "- Exponential, with rate $\\lambda=0.25$ (scale is the inverse)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(A.2) Display the histogram of the samples, along with the curve of the expression for the probability density function. Do they match?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Discrete random variables**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(B.1) Generate $N=1000$ samples using the following distributions:\n",
    "\n",
    "- Bernoulli, with $p=0.7$\n",
    "- Binomial, with $N_o=10$, $p=0.7$\n",
    "- Geometric, with $p=0.7$\n",
    "- Poisson, with $\\lambda = 7$\n",
    "- Binomial, with $N_o=1000$, $p=0.007$\n",
    "\n",
    "(*) compare the histograms for the last two cases. What do you observe?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(B.2) Display the histogram of the samples, along with the curve of the expression for the probability density function. Do they match?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2 (LLN)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are $N$ real numbers available, generated by independent identically distributed (i.i.d.) repetitions of the same experiment, so that each number is drawn from the same random distribution. Validate that the LLN holds true if the distribution is:\n",
    "\n",
    "a. $\\mathrm{Normal}$($\\mu,\\sigma^2$), $\\mu=2,\\ \\sigma=1$\n",
    "\n",
    "b. $\\mathrm{Normal}$($\\mu,\\sigma^2$), $\\mu=2,\\ \\sigma=30$\n",
    "\n",
    "c. $\\mathrm{Poisson}$($\\lambda$), $\\lambda = 10$\n",
    "\n",
    "> For numerical values, test the LLN using : <font color=blue>$N=10,\\ 100,\\ 1000,\\ 10000,\\ 100000,\\ 1000000$</font>.\n",
    "\n",
    "Plot the absolute error between theory and estimated value. Write a comment about your observations in all three cases.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 10\n",
    "L = list([10,100,1000,10000,100000,1000000])\n",
    "# Hint: use \n",
    "# np.random.normal, np.random.poisson, np.mean\n",
    "#TODO"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f9ec286d8d0>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the estimation error\n",
    "plt.plot(L,Ra,label='Normal(2,1)')\n",
    "plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('error')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f9ec49a21d0>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(L,Rb,label='Normal(2,900)')\n",
    "plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('error')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f9ec5ccc320>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(L,Rc,label='Poisson(10)')\n",
    "plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('error')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3 (CLT application 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A data-centre serves requests of service that arrive over time. Suppose the service time $X_i$ for request $i$ is exponentially distributed and has mean $E[X_i] = 1$ [msec] and variance $Var[X_i] = 1$ [msec]. Let $Y$ be the total time necessary to serve $n = 50$ requests. What is the probability that this time ranges between $40 ≤ Y ≤ 60$ [msec]?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer**\n",
    "\n",
    "From the solution on the slides we saw that \n",
    "\n",
    "$P(40\\leq Y\\leq 60) \\approx \\Phi(\\sqrt{2})-\\Phi(-\\sqrt{2})$. \n",
    "\n",
    "Here $\\Phi(x)$ is the CDF of the normal distribution for value $x$. The exact value of the above can be derived from existing tables found in Python."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math \n",
    "# Hint: use norm.cdf\n",
    "# TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see how good this approximation is, by generating a large number of normal vectors, each with $50$ entries and averaging. Let us see the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 50 # size of each data-set\n",
    "It = 10000 # number of iterations\n",
    "S = list() # list of different sums_50\n",
    "U = list() # list of indicators\n",
    "\n",
    "for i in range(0,It):\n",
    "    #TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4 (CLT application 2) - bonus*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A packet consists of $n=1000$ [bits]. Due to noice, bit errors occur independently, with probability $p=0.1$. What is the probability to have more than $120$ errors within the same packet?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer**\n",
    "We see that each bit can be correct or in error with probability $0.9$ and $0.1$ respectively. As a first approach we can see that the repetition of $1000$ Bernoulli experiments corresponds to a binomial distribution $Z$. We can then write that \n",
    "\n",
    "$P(Z > 120) = 1-P(Z\\leq 120) =$ $\\sum_{x=0}^{120} \\frac{1000!}{x!(1000-x)!}0.1^x0.9^{1000-x}$. \n",
    "\n",
    "The exact value can be found by calculating the exact sum.\n",
    "\n",
    "TODO"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.special as spec\n",
    "# Hint: use spec.binom in the calculation of the above sum\n",
    "# TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we proceed with the CLT approximation. The answer is given in the slides:  TODO"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Hint: use norm.cdf\n",
    "# TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The END"
   ]
  }
 ],
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   "language": "python",
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    "name": "ipython",
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