{ "cells": [ { "cell_type": "markdown", "id": "c57b2ce5", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Distributions and Random Numbers\n", "\n", "\n", "Feng Li\n", "\n", "School of Statistics and Mathematics\n", "\n", "Central University of Finance and Economics\n", "\n", "[feng.li@cufe.edu.cn](mailto:feng.li@cufe.edu.cn)\n", "\n", "[https://feng.li/statcomp](https://feng.li/statcomp)\n" ] }, { "cell_type": "markdown", "id": "ed515448", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Basic concepts of random numbers\n", "\n", "### Preliminary\n", "\n", "- **Pseudo random numbers**\n", "\n", " - an algorithm for generating a sequence of numbers that\n", " approximates the properties of random numbers.\n", "\n", " - The sequence is not truly random in that it is completely\n", " determined by a relatively small set of initial values, called\n", " the PRNG’s state.\n", "\n", " - Pseudo random numbers are important in practice for their\n", " **speed** in number generation and their **reproducibility**." ] }, { "cell_type": "markdown", "id": "b6a6c8d8", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- **Random seed**\n", "\n", " A random seed (or seed state, or just seed) is a number (or vector)\n", " used to initialize a pseudo random number generator.\n", "\n", "- The most important random numbers are from uniform distributed\n", " numbers.\n", "\n", " > runif(n,a,b)\n", "\n", "- Numbers selected from a non-uniform probability distribution can be\n", " generated using a uniform distribution PRNG and a function that\n", " relates the two distributions.\n", "\n", "- Assume you have uniformly distributed random numbers from \$0, 1\$,\n", " how do you extend it to \$a, b\$?" ] }, { "cell_type": "markdown", "id": "88049805", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Continuous random variables\n", "\n", "### Normal Distribution\n", "\n", "- **The normal density function**\n", " $$f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2} }$$\n", "\n", " > dnorm(x,mu,sigma)\n", "\n", " > dnorm(x,mu,sigma, log=TRUE)\n", "\n", " - In theory, dnorm(x,mu,sigma, log=TRUE)==log(dnorm(x,mu,sigma))\n", " \n", " - but dnorm(x,mu,sigma, log=TRUE) but is more stable for very\n", " large values. Why?\n", "\n", " - **We love logs**." ] }, { "cell_type": "code", "execution_count": 6, "id": "34388c58", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "text/html": [ "0" ], "text/latex": [ "0" ], "text/markdown": [ "0" ], "text/plain": [ "[1] 0" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "-Inf" ], "text/latex": [ "-Inf" ], "text/markdown": [ "-Inf" ], "text/plain": [ "[1] -Inf" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "-5000.9189385332" ], "text/latex": [ "-5000.9189385332" ], "text/markdown": [ "-5000.9189385332" ], "text/plain": [ "[1] -5000.919" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "dnorm(100, mean=0, sd=1)\n", "log(dnorm(100, mean=0, sd=1))\n", "\n", "dnorm(100, mean=0, sd=1, log=TRUE)" ] }, { "cell_type": "markdown", "id": "3dbf69b5", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- **The CDF** (Cumulative Distribution Function)\n", " $$\\Phi(x)\\;=\\int_{-\\infty}^x f(t, \\mu, \\sigma) d t$$\n", "\n", " > pnorm(q,mu,sigma)\n", "\n", "- **The quantile** (Given CDF, what is x?), i.e. $\\Phi^{-1}(p)$\n", "\n", " > qnorm(p,mu,sigma)\n", "\n", "- **Random numbers from normal distribution**\n", "\n", " > rnorm(n,mu,sigma)" ] }, { "cell_type": "code", "execution_count": 5, "id": "ea776f8c", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Plot with title “Histogram of z”" ] }, "metadata": { "image/png": { "height": 180, "width": 240 } }, "output_type": "display_data" } ], "source": [ "z = rnorm (100,mean=0, sd=1)\n", "hist(z)" ] }, { "cell_type": "code", "execution_count": 3, "id": "de109e18", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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UcfYdgwMyajFnJyQlwcpkzBrFnq2V+rpaUhLQ1t22LMGDzzDEaORMeOZk9JDRJ9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\nZYpTAAAAAElFTkSuQmCC", "text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 180, "width": 240 } }, "output_type": "display_data" } ], "source": [ "x = seq(-5, 5, 0.1)\n", "y = dnorm(x = x, mean = 0, sd = 1)\n", "plot(x, y, col = \"blue\", type = \"l\", lwd = 4)" ] }, { "cell_type": "code", "execution_count": 4, "id": "262ed398", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 180, "width": 240 } }, "output_type": "display_data" } ], "source": [ "y1 = pnorm(q = x, mean = 0, sd = 1)\n", "plot(x, y1, col = \"blue\", type = \"l\", lwd = 4)" ] }, { "cell_type": "markdown", "id": "6b999182", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Other types of continuous distribution\n", "\n", "- Distribution Function in R\n", "\n", " Student t: {p,d,q}t\n", " \n", " Chi squared: {p,d,q}chi\n", " \n", " Gamma: {p,d,q}gamma\n", " \n", " Exponential {p,d,q}exp\n", "\n", "\n", "- For a significance test, what distribution do you use?\n" ] }, { "cell_type": "markdown", "id": "f99866f8", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Discrete random variables\n", "\n", "### Discrete random variables\n", "\n", "- Distribution Function in R\n", "\n", " Binomial: {p,d,q}binorm\n", " \n", " Negative binomial: {p,d,q}nbinom\n", "\n", " Poisson: {p,d,q}pois\n", " \n", " Geometric: {p,d,q}geom\n", " \n", "- Bernoulli distribution is a special case of binomial distribution." ] }, { "cell_type": "markdown", "id": "2c3bbedf", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Suggested reading\n", "\n", "- Jones (2009): **Chapter 14, 15, 16**" ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "R", "language": "R", "name": "ir" }, "language_info": { "codemirror_mode": "r", "file_extension": ".r", "mimetype": "text/x-r-source", "name": "R", "pygments_lexer": "r", "version": "4.3.2" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": false, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 5 }