{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Rozdelenia v spracovaní chýb\n", "Hustoty a ich číselné a grafické charakteristiky" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Rozdelenia v Geogebre\n", "https://www.geogebra.org/m/kySmQVMn#material/hFV8Gz4P\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Rovnomerné rozdelenie\n", "\n", "Hustota rozdelenia \n", "\n", "$f(x) = \\begin{cases}\n", " \\displaystyle \\frac{1}{a} & \\text{pre } -a/2 \\le x\\le a/2 \\\\ \n", " 0 & \\text{pre } x <-a/2 \\,\\text{ alebo } x > a/2\n", " \\end{cases}$" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# hustota rozdelenia\n", "var('a')\n", "assume(a>0)\n", "f(x) = 1/a\n", "f(x).show()" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "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" }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# graf\n", "plot(f(x,a=1),-1/2,1/2)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1" ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# pravdepodobnosť - plocha pod grafom hustoty\n", "p = integral(f(x),x,-a/2,a/2)\n", "p" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# stredná hodnota\n", "m = integral(x*f(x),x,-a/2,a/2)\n", "m" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# rozptyl\n", "s2 = integral(x^2*f(x),x,-a/2,a/2)\n", "s2.show()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# štandardná neistota\n", "u=sqrt(s2)\n", "u.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Trojuholníkové rozdelenie\n", "\n", "Hustota rozdelenia \n", "\n", "$f(x) = \\begin{cases}\n", " \\dfrac{2}{a} \\left(1-\\dfrac{2}{a}x \\, \\mathrm{sgn}\\, x \\right) & \\text{pre } -a/2 \\le x\\le a/2 \\\\ \n", " 0 & \\text{pre } x < -a/2 \\text{ alebo } x > a/2\n", " \\end{cases}$" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# hustota\n", "var('a')\n", "assume(a>0)\n", "f(x) = 2/a*(1-2/a*x*sgn(x))\n", "f(x).show()" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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}, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# graf\n", "plot(f(x,a=1),-1/2,1/2)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1" ] }, "execution_count": 9, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# pravdepodobnosť pod grafom hustoty\n", "p = integral(f(x),x,-a/2,a/2)\n", "p" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# stredná hodnota\n", "m = integral(x*f(x),x,-a/2,a/2)\n", "m" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# rozptyl\n", "s2 = integral(x^2*f(x),x,-a/2,a/2)\n", "s2.show()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "# štandardná neistota\n", "u = sqrt(s2)\n", "u.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Normálne rozdelenie\n", "\n", "Hustota rozdelenia \n", "\n", "$f(x) = \\sqrt{ \\dfrac{e^{-x^2/s^2}}{2\\pi s^2}}$ alebo $f(x) = \\dfrac{1}{\\sqrt{2\\pi}s}e^{-\\frac{x^2}{2s^2}}$ \n", "\n", "pričom $ s = a/6$\n" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "# hustota\n", "var('s')\n", "assume(s>0)\n", "f(x) = 1/(s*sqrt(2*pi))*exp(-x^2/(2*s^2))\n", "show(f(x))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "# graf\n", "plot(f(x,s=1/6),-1/2,1/2)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "# pravdepodobnosť pod grafom hustoty\n", "p = integral(f(x),x,-oo,oo)\n", "p.show()\n", "p.simplify().show()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "# stredná hodnota\n", "m = integral(x*f(x),x,-oo,oo)\n", "m" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "# rozptyl\n", "m = integral(x^2*f(x),x,-oo,oo)\n", "m.simplify().show()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ ], "source": [ "# štandardná neistota\n", "u = sqrt(m.simplify())\n", "u.show()" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath (stable)", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 0 }