% \documentclass[serif]{beamer} % Serif for Computer Modern math font.
\documentclass[serif, handout]{beamer} % Handout to ignore pause statements.
\hypersetup{colorlinks,linkcolor=,urlcolor=red}
\usefonttheme{serif} % Looks like Computer Modern for non-math text -- nice!
\setbeamertemplate{navigation symbols}{} % Suppress navigation symbols
% \usetheme{Berlin} % Displays sections on top
\usetheme{Frankfurt} % Displays section titles on top: Fairly thin but still swallows some material at bottom of crowded slides
%\usetheme{Berkeley}
\usepackage[english]{babel}
\usepackage{amsmath} % for binom
\usepackage{amsfonts} % for \mathbb{R} The set of reals
% \usepackage{graphicx} % To include pdf files!
% \definecolor{links}{HTML}{2A1B81}
% \definecolor{links}{red}
\setbeamertemplate{footline}[frame number]
\mode
\title{Continuous Random Variables\footnote{ This slide show is an open-source document. See last slide for copyright information.}\\(Section 2.4 and parts of 2.5)}
\subtitle{STA 256: Fall 2019}
\date{} % To suppress date
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{Overview}
\tableofcontents
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Continuous Random Variables}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Formal Definitions}
%\framesubtitle{}
\begin{itemize}
\item Our textbook makes a distinction between continuous random variables and absolutely continuous random variables.
\item All absolutely continuous random variables are continuous.
\item There are continuous random variables that are not absolutely continuous.
\item But the examples are too advanced for us right now. \pause
\item Book says (p.~53) ``In fact, statisticians sometimes say that $X$ is continuous as shorthand for saying that $X$ is absolutely continuous."
\item That is what we will do.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Continuous Random Variables: The idea}
\framesubtitle{Probability is area under a curve} \pause
%{\small
\begin{itemize}
\item Discrete random variables take on a finite or countably infinite number of values. \pause
\item Continuous random variables take on an \emph{uncountably infinite} number of values. \pause
\item This implies that $S$ is uncountable too, but we seldom talk about it. \pause
\item Probability is area under a curve \pause --- that is, area between a curve and the $x$ axis. \pause
\begin{center}
\includegraphics[width=3in]{area}
\end{center} % \pause
% Maybe end with this.
% \item Nature may be fundamentally discrete. \pause
% \item If so, continuous probability is a convenient approximation.
\end{itemize}
%} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Probability Density Function} \pause
%\framesubtitle{}
\begin{center}
\includegraphics[width=3in]{area}
\end{center} \pause
\begin{displaymath}
P(a 1$, \pause
\begin{eqnarray*}
F(x) &=& \int_{-\infty}^0 0 \, dt + \int_0^1 2t \, dt
+\int_1^x 0 \, dt \\ \pause
&=& 0 + 1 + 0 \\ \pause
&=& 1
\end{eqnarray*}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Putting the pieces together} \pause
%\framesubtitle{}
{\LARGE
\begin{displaymath}
F(x) = \left\{ \begin{array}{ll} % ll means left left
0 & \mbox{for $x < 0$} \\
x^2 & \mbox{for } 0 \leq x \leq 1 \\
1 & \mbox{for } x > 1
\end{array} \right. % Need that crazy invisible right period!
\end{displaymath} \pause
} % End size
\vspace{10mm}
The derivation does not need to be this detailed, but the final result has to be complete. More examples will be given.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Common Continuous Distributions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Common Continuous Distributions}
%\framesubtitle{}
\begin{itemize}
\item Uniform
\item Exponential
\item Gamma
\item Normal
\item Beta
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Uniform Distribution: $X\sim$ Uniform$(L,R)$}
\framesubtitle{Parameters $L 0$} % Example 2.4.5
\begin{columns}
\column{0.5\textwidth}
{\large
\begin{displaymath}
f(x) = \left\{ \begin{array}{ll} % ll means left left
\lambda e^{-\lambda x} & \mbox{for $x \geq 0$} \\
0 & \mbox{for } x < 0
\end{array} \right. % Need that crazy invisible right period!
\end{displaymath} \pause
} % End size
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2.4in]{Exponential}
\end{center}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Gamma Distribution: $X\sim$ Gamma($\alpha,\lambda$)}
\framesubtitle{Parameters $\alpha>0$ and $\lambda > 0$} % Example 2.4.6
\begin{columns}
\column{0.5\textwidth}
{\small
\begin{displaymath}
f(x) = \left\{ \begin{array}{ll} % ll means left left
\frac{\lambda^\alpha}{\Gamma(\alpha)} e^{-\lambda x} \, x^{\alpha-1} & \mbox{for $x \geq 0$} \\
0 & \mbox{for } x < 0
\end{array} \right. % Need that crazy invisible right period!
\end{displaymath} \pause
} % End size
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2.2in]{Gamma}
\end{center} \pause
\end{columns} \vspace{1mm}
{\small
The gamma function is defined by $\Gamma(\alpha) = \int_0^\infty e^{-t} \, t^{\alpha-1} \, dt$
\pause \vspace{2mm}
Integration by parts shows $\Gamma(\alpha+1) = \alpha \, \Gamma(\alpha)$.
} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Normal Distribution: $X\sim$ N($\mu,\sigma^2$)}
\framesubtitle{Parameters $\mu \in \mathbb{R}$ and $\sigma > 0$} % Example 2.4.8
\begin{columns}
\column{0.5\textwidth}
%{\large
\begin{eqnarray*}
f(x) & = & \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \\ \pause
& = & \frac{1}{\sigma \sqrt{2\pi}}\exp - \left\{{\frac{(x-\mu)^2}{2\sigma^2}}\right\}
\end{eqnarray*} \pause
%} % End size
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2.1in]{Normal}
\end{center} \pause
\end{columns} \vspace{1mm}
The normal distribution is also called the Gaussian, or the ``bell curve." if $\mu=0$ and $\sigma=1$, we write $X\sim$ N(0,1) and call it the \emph{standard normal}.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The book uses parameters a, b. But it's only in a HW problem anyway, and I can't stand to re-do all the pictures, where I worked hard to get an alpha and a beta.
\begin{frame}
\frametitle{The Beta Distribution: $X\sim$ Beta($\alpha,\beta$)}
\framesubtitle{Parameters $\alpha>0$ and $\beta > 0$} % HW problem 2.4.24
{\Large
\begin{displaymath}
f(x) = \left\{ \begin{array}{ll} % ll means left left
\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \,
x^{\alpha-1} (1-x)^{\beta-1} & \mbox{for $0 \leq x \leq 1$} \\
0 & \mbox{Otherwise}
\end{array} \right. % Need that crazy invisible right period!
\end{displaymath} \pause \vspace{3mm}
} % End size
\vspace{3mm}
Using $\Gamma(n+1) = n \, \Gamma(n)$ and $\Gamma(\beta) = \int_0^\infty e^{-t} \, t^{\beta-1} \, dt$, note that a beta distribution with $\alpha=\beta=1$ is Uniform(0,1). \pause \vspace{2mm}
The beta density can assume a variety of shapes, depending on the parameters $\alpha$ and $\beta$.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Beta density with $\alpha=5$ and $\beta=5$}
%\framesubtitle{}
\begin{center}
\includegraphics[width=3.1in]{Beta55}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Beta density with $\alpha=8$ and $\beta=2$}
%\framesubtitle{}
\begin{center}
\includegraphics[width=3.1in]{Beta82}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Beta density with $\alpha=2$ and $\beta=8$}
%\framesubtitle{}
\begin{center}
\includegraphics[width=3.1in]{Beta28}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Beta density with $\alpha=\frac{1}{2}$ and $\beta=\frac{1}{2}$}
%\framesubtitle{}
\begin{center}
\includegraphics[width=3.1in]{BetaHalfHalf}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Copyright Information}
This slide show was prepared by \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
{Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\vspace{5mm}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$ = \{\omega \in \Omega: \}$
\begin{frame}
\frametitle{} \pause
%\framesubtitle{}
\begin{itemize}
\item \pause
\item \pause
\item
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
R work for density plots
rm(list=ls())
# Uniform
tstring = 'Uniform(L,R) density with L=0 and R=1'
x = seq(from=0,to=1,by=0.01); Density = x-x + 1
plot(x,Density,type = 'l',main=tstring)
# Exponential
tstring = expression(paste('Exponential(',lambda,') density with ',lambda,'= 1'))
x = seq(from=0,to=4,by=0.05); Density = exp(-x)
plot(x,Density,type = 'l',main=tstring)
# Gamma
tstring = expression(paste('Gamma(',alpha,',',lambda,') density with ',
alpha,' = 3 and ',lambda,'= 1'))
x = seq(from=0,to=10,by=0.05); Density = dgamma(x,shape=3,rate=1)
plot(x,Density,type = 'l',main=tstring)
# Normal
tstring = expression(paste('Normal(',mu,',',sigma,') density with ',
mu,' = 0 and ',sigma,'= 1'))
x = seq(from=-4,to=4,by=0.05); Density = dnorm(x)
plot(x,Density,type = 'l',main=tstring)
# Beta
tstring = expression(paste('Beta(',alpha,',',beta,') density with ',
alpha,' = 5 and ',beta,'= 5'))
x = seq(from=0,to=1,by=0.01); Density = dbeta(x,5,5)
plot(x,Density,type = 'l',main=tstring)
% Put this stuff in a separate slide show.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Functions of a Random Variable}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Functions of a Random Variable} \pause
%\framesubtitle{}
\begin{itemize}
\item Suppose you know the probability distribution of $X$. \pause
\item $Y=g(X)$. \pause
\item What is the probability distribution of $Y$? \pause
\item For example, $X$ is miles per gallon. \pause
\item $Y$ is litres per 100 kilometers for the same population of cars. \pause
\item You can make probability statements about $X$, but you need to make probability statements about $Y$
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{General approach to finding the density of $Y=g(X)$} \pause
%\framesubtitle{}
First, find the set of $y$ values where $f_y(y)>0$. \pause %\vspace{3mm}
\begin{itemize}
\item There are infinitely many right answers, differing only on a set of probability zero. \pause
\item If $f_x(x)>0$, let $f_y(y)>0$ for $y=g(x)$. \pause This works. \pause
% Expected value of indicator, change of variables theorem.
\item Then, for \emph{one of those $y$ values} (assuming $f_y(y)$ continuous there), \pause
\begin{displaymath}
f_y(y) = \frac{d}{dy} \, F_y(y) \pause
= \frac{d}{dy} \, P(Y \leq y)
\end{displaymath} \pause
\item Substitute for $Y$ in terms of $X$. \pause
\item Try to solve for $X$. \pause
\item Express in terms of the cdf $F_x(x)$. \pause
\item Differentiate with respect to $y$. \pause
\item Usually use the chain rule. \pause
\item We need an example.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example}
%\framesubtitle{}
Let $X\sim$ Gamma($\alpha,\lambda$) \pause and $Y = 2X$. \pause Find the density of $Y$. \pause
\begin{displaymath}
f_x(x) = \left\{ \begin{array}{ll} % ll means left left
\frac{\lambda^\alpha}{\Gamma(\alpha)}e^{-\lambda x} \, x^{\alpha-1}
& \mbox{for $x \geq 0$} \\
0 & \mbox{for } x < 0
\end{array} \right. % Need that crazy invisible right period!
\end{displaymath} \pause
First, where will $f_y(y)$ be non-zero? \pause
\begin{itemize}
\item $f_x(x)>0$ for $x \geq 0$. \pause
\item $x \geq 0 \pause \Leftrightarrow y=2x \geq 0$. \pause
\item So, for $y \geq 0$, \ldots
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Derive the functional part of $f_y(y)$}
\framesubtitle{$X\sim$ Gamma($\alpha,\lambda$) and $Y = 2X$} \pause
\begin{columns}
\column{0.4\textwidth}
\begin{eqnarray*}
f_y(y) & = & \frac{d}{dy} \, F_y(y) \\ \pause
& = & \frac{d}{dy} \, P(Y \leq y) \\ \pause
& = & \frac{d}{dy} \, P(2X \leq y) \\ \pause
& = & \frac{d}{dy} \, P\left(X \leq \frac{1}{2}y\right) \\ \pause
& = & \frac{d}{dy} \, F_x\left(\frac{1}{2}y\right) \\ \pause
\end{eqnarray*}
\column{0.6\textwidth}
\begin{eqnarray*}
& = & f_x\left(\frac{1}{2}y\right) \pause \cdot \pause \frac{d}{dy} \frac{1}{2}y\\ \pause
& = & \frac{1}{2} \cdot \pause \frac{\lambda^\alpha}{\Gamma(\alpha)}
\exp\left\{-\lambda \frac{1}{2}y\right\} \, \left(\frac{1}{2}y\right)^{\alpha-1}
\\ \pause
& = & \frac{(\lambda/2)^\alpha}{\Gamma(\alpha)}
\exp\left\{- \frac{\lambda}{2}y\right\} \, y^{\alpha-1} \\ \pause
&& \\ % Put blank lines lines to align columns.
&& \\
&& \\
&& \\
\end{eqnarray*}
\end{columns}
Compare gamma density:
$f_x(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)}e^{-\lambda x} \, x^{\alpha-1}$ for $x \geq 0$.
\pause
Conclude $Y\sim$ Gamma($\alpha,\lambda/2$).
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Give the density of $Y$}
\framesubtitle{Don't forget to specify where $f_y(y)>0$} \pause
{\LARGE
\begin{displaymath}
f_y(y) = \left\{ \begin{array}{ll} % ll means left left
\frac{(\lambda/2)^\alpha}{\Gamma(\alpha)}
\exp\left\{- \frac{\lambda}{2}y\right\} \, y^{\alpha-1},
& \mbox{for $y \geq 0$} \\
0 & \mbox{for } y < 0
\end{array} \right. % Need that crazy invisible right period!
\end{displaymath}
} % End size
\end{frame}