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{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Random variable\n", "A random variable $\\Chi$ is a function\n", "\n", "$X: \\quad \\Omega \\to W_{\\Chi} \\subset \\mathbb{R}$\n", "\n", "$\\phantom{\\Chi:} \\quad \\omega \\mapsto \\Chi(\\omega)$\n", "\n", "where $W_{\\Chi}$ is called range of the random variable ${\\Chi}$\n", "\n", "### Functions\n", "Functions of the type $\\Omega \\mapsto \\mathbb{R}$ can be added:\n", "\n", "$(\\Chi + \\gamma)(\\omega) = \\Chi(\\omega) + \\gamma(\\omega), \\quad \\forall \\omega \\in \\Omega$\n", "\n", "$(\\Chi \\cdot \\gamma)(\\omega) = \\Chi(\\omega) \\cdot \\gamma(\\omega), \\quad \\forall \\omega \\in \\Omega$\n", "\n", "### Probability\n", "\n", "The probability that $\\Chi$ takes the value $x$ is calculated as follow:\n", "\n", "$\\displaystyle P(\\Chi = x) = P(\\{\\omega | \\Chi(\\omega)\\}) = \\sum_{\\omega;\\Chi(\\omega)=x} P(\\omega)$\n", "\n", "## Probability distribution\n", "The probability distribution of a random variable $\\Chi$ is the specification of the probability of the event $\\Chi = x$ for each realization $x \\in W_{\\Chi}$:\n", "\n", "$x \\in W_{\\Chi} = P(\\Chi = x)$\n", "\n", "### Distribution identity\n", "For the probability distribution of a discrete random variable $\\Chi$, the sum of the $P(\\Chi = x)$ over all possible realizations $x$ equals one:\n", "\n", "$\\displaystyle \\sum_{x \\in W_{\\Chi}} P(\\Chi = x) = 1$\n", "\n", "## Expected value\n", "Let $\\Chi$ be a discrete random variable $\\Chi$. The expected value $E(\\Chi) \\in \\mathbb{R}$ of $\\Chi$ is defined as:\n", "\n", "$\\displaystyle E(\\Chi) = \\sum_{x \\in W_{\\Chi}} xP(\\Chi = x), \\:$ where $W_{\\Chi} :$ range of $\\Chi$.\n", "\n", "## Variance and Standard Deviation\n", "The variance $\\text{Var}(\\Chi) \\in \\mathbb{R}$ and the standard deviation $\\sigma(\\Chi)$ of $\\Chi$ are defined as:\n", "\n", "$\\text{Var}(\\Chi) \\displaystyle = \\sum_{x \\in W_{\\Chi}} (x - E(\\Chi))^2P(\\Chi = x)$ \n", "\n", "$\\displaystyle \\sigma(\\Chi) = \\sqrt{\\text{Var}(\\Chi)}$\n", "\n", "For the variance of a random variable $\\Chi$, we have\n", "\n", "$\\text{Var}(\\Chi) = E((\\Chi - E(\\Chi))^2)$\n", "\n", "## Cumulative distribution function\n", "The cumulative distribution function of a random variable $\\Chi$ is the function:\n", "\n", "$\\displaystyle F(x) = P(\\Chi \\leq x) = \\sum_{y\\leq x} P(\\Chi = y)$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "<Figure size 640x480 with 1 Axes>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "values_jass=[-1,0,2,3,4,10,11,12]\n", "prob_jass=np.array([0,4/9,1/9,1/9,1/9,1/9,1/9,0])\n", "cum_jass=np.cumsum(prob_jass)\n", "\n", "plt.step(values_jass,cum_jass, where='post')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Binomial distribution\n", "\n", "### Bernoulli distribution\n", "The distribution of a random variable $\\Chi$ with values $W = \\{0,1\\}$ can be described by a single parameter $p$:\n", "\n", "$P(\\Chi = 1) = p, \\qquad P(\\Chi = 0) = 1 - p \\qquad 0 \\leq p \\leq 1$\n", "\n", "This distribution is called the Bernoulli distribution, and we write it as:\n", "\n", "$\\Chi \\sim Bernoulli(p)$\n", "\n", "### Binomial $(n,p)$ distribution\n", "A random variable $\\Chi$ with values in $\\{0,1,2,...,n\\}$ is said to be binomially distributed with parameters $n$ and $p$ if:\n", "\n", "$\\displaystyle P(\\Chi = x) = {\\binom n x} p^x(1-p)^{n-x},\\ \\forall x \\in \\{0,1,2,...,n\\}$\n", "\n", "Here, $0 \\leq p \\leq 1$ is the success parameter of the distribution. We also write:\n", "\n", "$\\displaystyle \\Chi \\sim {\\binom n p}$ or $\\Chi \\sim \\text{Bin}(n,p)$\n", "\n", "## Characteristics of the binomial distribution\n", "Let $\\Chi \\sim \\text{Bin}(n,p)$. Then the expected value of $\\Chi$ is:\n", "\n", "$E(\\Chi) = np$\n", "\n", "The variance of $\\Chi$ is:\n", "\n", "$\\text{Var}(\\Chi) = np(1-p)$\n", "\n", "and finally the standard deviation:\n", "\n", "$\\sigma(\\Chi) = \\sqrt{\\text{Var}(\\Chi)}$" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" } }, "nbformat": 4, "nbformat_minor": 2 }
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