lab-sessions-26a/session-6-distributions/main-session.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "5d24fc75",
   "metadata": {},
   "outputs": [],
   "source": [
    "# General used libriries\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import binom, poisson, norm, expon "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f8545a0",
   "metadata": {},
   "source": [
    "# Bernoulli distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "38c1af9c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(X=1) =  0.7\n",
      "P(X=0) =  0.30000000000000004\n",
      "Mean =  0.7\n",
      "Variance =  0.21000000000000002\n"
     ]
    }
   ],
   "source": [
    "# Bernoulli distribution:\n",
    "p = 0.7\n",
    "print(\"P(X=1) = \", p)\n",
    "print(\"P(X=0) = \", 1-p)\n",
    "print(\"Mean = \", p)\n",
    "print(\"Variance = \", p*(1-p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "2ab9219a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First 20 observations:\n",
      "[1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1]\n",
      "Sample mean:  0.675\n",
      "Sample variance:  0.21959459459459454\n"
     ]
    }
   ],
   "source": [
    "# Bernoulli observations \n",
    "# repeating a success|failures experiment many times\n",
    "p = 0.7\n",
    "n = 1000 #samples\n",
    "np.random.seed(33)\n",
    "berData = np.random.binomial(1, p, size=n)\n",
    "print(\"First 20 observations:\")\n",
    "print(berData[:20])\n",
    "\n",
    "print(\"Sample mean: \", berData.mean())\n",
    "print(\"Sample variance: \", berData.var(ddof=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e2ae2434",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "values, counts = np.unique(berData, return_counts=True)\n",
    "plt.bar(values, counts/n, width=0.4, color=\"grey\")\n",
    "plt.xticks([0, 1])\n",
    "plt.xlabel(\"Outcomes\")\n",
    "plt.ylabel(\"Relative frequency\")\n",
    "plt.title(\"Simulated Bernoulli Distribution\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98f319fd",
   "metadata": {},
   "source": [
    "# Binomial distribution\n",
    "The binomial distribution counts the number of successes in $n$ independent Bernoulli trials."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "974eccf3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(X=2) = 0.2711\n",
      "Mean=  1.6\n",
      "Variance=  1.4720000000000002\n"
     ]
    }
   ],
   "source": [
    "p = 0.08\n",
    "n = 20\n",
    "k = 2\n",
    "\n",
    "prob = binom.pmf(k, n, p)\n",
    "print(f\"P(X=2) = {prob:.4f}\")\n",
    "print(\"Mean= \", n*p)\n",
    "print(\"Variance= \", n*p*(1-p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "87d14367",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(0, n+1)\n",
    "pmf = binom.pmf(x, n, p)\n",
    "plt.bar(x, pmf, color=\"grey\", width=0.6)\n",
    "plt.xlabel(\"Number of succeses\")\n",
    "plt.ylabel(\"Probability\")\n",
    "plt.title(\"Binomial distribution: n=20, p=0.08\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9c8de1d2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Simulated mean:  1.6011\n",
      "Theoretical mean:  1.6\n",
      "--------------\n",
      "Simulated var:  1.4717259625962598\n",
      "Theoretical var:  1.4720000000000002\n"
     ]
    }
   ],
   "source": [
    "# Binomial simuation\n",
    "n = 20 #sample\n",
    "p = 0.08\n",
    "nPop = 10_000\n",
    "np.random.seed(33)\n",
    "binomData = np.random.binomial(n,p,size=nPop)\n",
    "\n",
    "print(\"Simulated mean: \", binomData.mean())\n",
    "print(\"Theoretical mean: \", n*p)\n",
    "print(\"--------------\")\n",
    "print(\"Simulated var: \", binomData.var(ddof=1))\n",
    "print(\"Theoretical var: \", n*p*(1-p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "ec289f08",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(binomData, bins=np.arange(-0.5,n+1.5,1), density=True, color=\"grey\")\n",
    "plt.plot(x, pmf, 'ok')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "5ce9af51",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.88693329e-01 3.28162312e-01 2.71090605e-01 1.41438577e-01\n",
      " 5.22707783e-02 1.45449122e-02 3.16193744e-03 5.49902164e-04\n",
      " 7.77035666e-05 9.00910917e-06 8.61740877e-07 6.81218085e-08\n",
      " 4.44272664e-09 2.37737880e-10 1.03364296e-11 3.59527985e-13\n",
      " 9.76978221e-15 1.99893242e-16 2.89700351e-18 2.65171946e-20\n",
      " 1.15292150e-22]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.32816231158747255"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(pmf)\n",
    "20*0.08*(1-0.08)**(19)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2612a6fd",
   "metadata": {},
   "source": [
    "# Poisson distribution\n",
    "- $P(X=k) = e^{-\\lambda} \\lambda^k/k!$\n",
    "- $E[X] = \\lambda$\n",
    "- $Var(X) = \\lambda$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e94cffe1",
   "metadata": {},
   "source": [
    "Example: Let $X$ be the number of flaws per metre per cable, with average rate $\\lambda=3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "c629df8b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(X=0) =  0.049787068367863944\n",
      "P(X=2) =  0.22404180765538775\n",
      "Mean =  3\n",
      "Var =  3\n"
     ]
    }
   ],
   "source": [
    "lam = 3\n",
    "print(\"P(X=0) = \", poisson.pmf(0, lam))\n",
    "print(\"P(X=2) = \", poisson.pmf(2, lam))\n",
    "print(\"Mean = \", lam)\n",
    "print(\"Var = \", lam)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "e8882be4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(0, 15)\n",
    "pmfPoisson = poisson.pmf(x, lam)\n",
    "\n",
    "plt.bar(x, pmfPoisson, width=0.7)\n",
    "plt.xlabel(\"Count\")\n",
    "plt.ylabel(\"Probability\")\n",
    "plt.title(r\"Poisson distribution: $\\lambda=3$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "dd34877c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Simulated mean:  2.9953\n",
      "Theoretical mean:  3\n",
      "Simulated Var:  2.9753754475447542\n",
      "Theoretical Var:  3\n"
     ]
    }
   ],
   "source": [
    "# Poisson distribution\n",
    "lam = 3\n",
    "samples = 10_000\n",
    "poiData = np.random.poisson(lam, size=samples)\n",
    "\n",
    "print(\"Simulated mean: \", poiData.mean())\n",
    "print(\"Theoretical mean: \", lam)\n",
    "\n",
    "print(\"Simulated Var: \", poiData.var(ddof=1))\n",
    "print(\"Theoretical Var: \", lam)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "abcf172f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(poiData, bins=np.arange(-0.5, 15.5,1), density=True)\n",
    "plt.plot(x, poisson.pmf(x, lam),'or')\n",
    "plt.xlabel(\"Counts\")\n",
    "plt.ylabel(\"Relative Freq/Probability\")\n",
    "plt.title(\"Poisson: Simulated vs Theory\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb795b6e",
   "metadata": {},
   "source": [
    "# Normal distribution\n",
    "If $X\\sim N(\\mu, \\sigma^2)$, then:\n",
    "- $\\mu$ is the mean,\n",
    "- $\\sigma$ is the standard deviation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0416787f",
   "metadata": {},
   "source": [
    "Example: Suppose a sensor output is a normally distributed with mean 50 and standard deviation 4. We compute the probability that the output is less than 58. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "ea59f656",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.9772498680518208)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu = 50 \n",
    "sigma = 4\n",
    "proba = norm.cdf(58, loc=mu, scale=sigma)\n",
    "proba"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "763cf868",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 800x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(mu - 4*sigma, mu + 4*sigma, 400)\n",
    "pdf = norm.pdf(x, loc=mu, scale=sigma)\n",
    "\n",
    "plt.figure(figsize=(8,4))\n",
    "plt.plot(x, pdf)\n",
    "plt.axvline(58, linestyle='--')\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Density\")\n",
    "plt.title(\"Normal Distribution\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "90c5658f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Simulated mean:  50.004935022325526\n",
      "Theoretical mean:  50\n",
      "Simulated std:  3.9780626491745017\n",
      "Theoretical std:  4\n"
     ]
    }
   ],
   "source": [
    "mu = 50\n",
    "sigma = 4 \n",
    "samples = 10_000\n",
    "\n",
    "normData = np.random.normal(loc=mu, scale=sigma, size=samples)\n",
    "print(\"Simulated mean: \", normData.mean())\n",
    "print(\"Theoretical mean: \", mu)\n",
    "print(\"Simulated std: \", normData.std(ddof=1))\n",
    "print(\"Theoretical std: \", sigma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "0b94b1c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(normData, bins=50, density=True)\n",
    "plt.plot(x, pdf, 'or')\n",
    "plt.xlabel(\"Value\")\n",
    "plt.ylabel(\"Density\")\n",
    "plt.title(\"Normal: Simulation vs Theory\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "995adcee",
   "metadata": {},
   "source": [
    "# Central Limit Theorem\n",
    "The central limit theorem (CLT) states that the sampling distribution of the sample mean becomes approximately normal as the sample size increases."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "b22b9a32",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "population = np.random.exponential(scale=2, size=100_000)\n",
    "plt.hist(population, bins=50, density=True)\n",
    "plt.xlabel(\"Value\")\n",
    "plt.ylabel(\"Density\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e1d70f56",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "39dad6b6",
   "metadata": {},
   "outputs": [],
   "source": [
    "def SampleMeans(population, sampleSize, nRep=5_000):\n",
    "    means = []\n",
    "    for _ in range(nRep):\n",
    "        sample = np.random.choice(population, size=sampleSize, replace=True)\n",
    "        means.append(sample.mean())\n",
    "        pass\n",
    "    return np.array(means)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "2e324399",
   "metadata": {},
   "outputs": [],
   "source": [
    "mean5 = SampleMeans(population, 5)\n",
    "mean30 = SampleMeans(population, 30)\n",
    "mean100 = SampleMeans(population, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "e96babca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(mean5, bins=40, density=True, color=\"orange\",  alpha=0.3)\n",
    "plt.hist(mean30, bins=40, density=True, color=\"green\",  alpha=0.3)\n",
    "plt.hist(mean100, bins=40, density=True, color=\"blue\",  alpha=0.3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "5a18e86e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Population mean:  1.9932073011758906\n",
      "Population var:  4.025192782387372\n",
      "Mean of 5 samples:  1.9998045611349438\n",
      "Var of 5 samples:  0.7837744844351563\n",
      "Mean of 30 samples:  1.992377755782367\n",
      "Var of 5 samples:  0.13206867208626297\n",
      "Mean of 100 samples:  1.994853961830046\n"
     ]
    }
   ],
   "source": [
    "# Numerical Check\n",
    "print(\"Population mean: \", population.mean())\n",
    "print(\"Population var: \", population.var())\n",
    "\n",
    "print(\"Mean of 5 samples: \", mean5.mean())\n",
    "print(\"Var of 5 samples: \", mean5.var(ddof=1))\n",
    "print(\"Mean of 30 samples: \", mean30.mean())\n",
    "print(\"Var of 5 samples: \", mean30.var(ddof=1))\n",
    "print(\"Mean of 100 samples: \", mean100.mean())"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": ".venv (3.14.3)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.14.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}