{ "cells": [ { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

Typová rozdělení
Diskrétní
Binomické
Poissonovo
Spojité
rovnoměrné rozdělení
exponenciální rozdělení
Breit-Wigner (Cauchy, Lorentz)
beta rozdělení (pro interval 0<x<1)
gama rozdělení (pro interval x>0)
χ2 rozdělení
Studentovo t-rozdělení (Gosset 1908)
Fischer-Snedecorovo rozdělení
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "from scipy import stats\n", "from matplotlib import pyplot as pl\n", "import numpy as np" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "Typová rozdělení\n", "===================\n", "\n", "Diskrétní\n", "--------------------\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Binomické\n", "\n", "výsledky opakovaných pokusů s náhodným jevem, který má 2 možné výsledky\n", "\n", "\\begin{equation} \\pi_n(r)= \\frac{n!}{p! (n-p)!} p^r (1-p)^{n-r} \\end{equation}\n", "\n", "__střední hodnota__ : $np$ \n", "__disperze__ : $np(1-p)$ \n", "__asymetrie__ : $\\frac{1-2p}{\\sqrt{np(1-p)}}$ \n", "__excess__ : $\\frac{1-6p(1-p)}{np(1-p)}$ \n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Poissonovo\n", "\n", "\\begin{equation} \\pi_n(r)= \\frac{1}{r!} \\mu^r \\mathrm{e}^{-\\mu} \\label{eq:pois} \\end{equation}\n", "\n", "__střední hodnota__ : $\\mu$ \n", "__disperze__ : $\\mu$ \n", "__asymetrie__ : $1/\\sqrt{\\mu}$ \n", "__excess__ : $1/\\mu$ \n", "\n", "limitně pro $\\mu \\to \\infty$ se blíží $N(\\mu,\\sqrt{\\mu})$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Spojité\n", "-----------------\n" ] }, { "cell_type": "markdown", "metadata": { "subvar": 1.23 }, "source": [ "### rovnoměrné rozdělení\n", "\n", "$$ f(x)=1/(b-a)\\ \\ldots\\ x \\in < a,b >$$\n", "\n", "__střední hodnota__ : $(a+b)/2$ \n", "__disperze__ : $(a-b)^2/12$ \n", "__asymetrie__ : 0 \n", "__excess__ : $-1.2$ \n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### exponenciální rozdělení\n", "\n", "$$ f(x)=\\exp(\\frac{-x}{\\mu})/\\mu $$\n", "\n", "__střední hodnota__ : $\\mu$ \n", "__disperze__ : $\\mu^2$ \n", "__asymetrie__ : 2 \n", "__excess__ : 6 \n", "\n", "charakter. funkce $1/(1-\\i\\mu t)$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Breit-Wigner (Cauchy, Lorentz)\n", "\n", "$$ f(x)=\\frac{1}{\\pi} \\frac{1}{1+(x-\\mu)^2} $$\n", "\n", "__střední hodnota__ : nedef. () \n", "__disperze__ : $\\infty$ \n", "\n", "charakter. funkce $\\exp^{-|t|}$\n", "\n", "*generování*: $\\tan \\pi(r-1/2)$ pro *r* s rovnom. rozdělením v (0,1)\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x=np.r_[0.01:3:0.02]\n", "pl.plot(x,1/np.pi/(1+(x-1.2)**2))\n", "pl.plot(x,0.5/np.pi/(0.25+(x-1.2)**2))\n", "pl.grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### beta rozdělení (pro interval 00)\n", "\n", "$$ f(x)=\\frac{\\mu^\\nu}{\\Gamma(\\nu)} x^{\\nu-1} \\exp(-x/\\mu) $$\n", "\n", "__střední hodnota__ : $\\nu/\\mu$ \n", "__disperze__ : $\\nu/\\mu^2$ \n", "__asymetrie__ : $2/\\sqrt{\\nu}$ \n", "__excess__ : $6/\\nu$ \n", "\n", "*generování* (pro $\\mu=1$): vezmeme *n+1* NP $r_i$ s rovnom. rozdělením v (0,1) a spočteme $x=- \\ln \\prod^{n+1}{r_i}$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $\\chi^2$ rozdělení\n", "\n", "rozdělení součtu čtverců $n$ NP s normálním rozdělením N(0,1)\n", "\n", "$$ f_n(x) = \\frac{x^{n/2-1} \\exp{(-x/2)}}{2\\Gamma(n/2)} $$\n", "\n", "__střední hodnota__ : $n$ \n", "__disperze__ : $2n$ \n", "__asymetrie__ : $\\sqrt{8/n}$ \n", "__excess__ : $12/n$ \n", "\n", "charakter. funkce $(1-2 \\i t)^{-n/2}$\n", "\n", "limitně pro $n \\to \\infty$ se blíží $N(n,2n)$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Studentovo t-rozdělení (Gosset 1908)\n", "\n", "rozdělení podílu nezávislých NP s normálním a $\\chi^2_n$ rozdělením\n", "\n", "$$ f_n(x) = \\frac{\\Gamma((n+1)/2)}{\\sqrt{n\\pi}\\Gamma(n/2)} \\left(1+\\frac{t^2}{n}\\right)^{-(n+1)/2}$$\n", "\n", "__střední hodnota__ : 0 \n", "__disperze__ : $n/(n-2)$ \n", "__asymetrie__ : 0 \n", "__excess__ : $6/(n-4)$ pro n>4 \n", "\n", "*generování*: dvě NP $r_1, r_2$ s rovnom. rozdělením v (0,1) , pokud $r_1<0.5$, $t=1/(4r_1-1)$, $v=r_2/t^2$, jinak $t=4r_1-3$,$v=r_2$. Hodnotu $x=t$ akceptujeme, pokud $v<1-|t|/2$ nebo $v<(1+t^2/n)^{-(n+1)/2}$ \n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Fischer-Snedecorovo rozdělení\n", "\n", "má podíl dvou náhodných proměnných s rozděleními $\\chi^2(n)/n$ a $\\chi^2(m)/m$ \n", "\n", "\n", "$$ f_{(n,m)}(x) = \\frac{m}{n}^{m/2} \\frac{\\Gamma((m+n)/2)}{\\Gamma(m/2) \\Gamma(n/2)} x^{(n-2)/2} \\left(1-\\frac{m}{n} x\\right)^{-(m+n)/2} $$\n", "\n", "__střední hodnota__ : $m/(m-2)$ \n", "__disperze__ : $\\frac{2n^2(m+n-2)}{m(n-2)^2(n-4)}$\n", "\n", "\n", "pro m=1 se redukuje na t-rozdělení \n", "\n", "$1/2 \\log F$ má pro velká m,n přibližně rozdělení $N((1/f_2- 1/f_1)/2, (1/f_2+1/f_1)/2)$, kde $f_1=m-1, f_2=n-1$ " ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 0.26904923, 3.71679186])" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Waxhv1nvToUOP9dwydrp9G/78E6q4fx1MlfkH5uP7pK8pCjuoeXzmzFG9dysv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\n+n4HLh7g7O2ztCjTwnUBOZi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