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\mode<presentation>

\title{Continuous Random Variables\footnote{ This slide show is an open-source document. See last slide for copyright information.}}
\subtitle{STA 256: Fall 2018}
\date{} % To suppress date

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Continuous Random Variables}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\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 $\Omega$ 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<X<b) = \int_a^b f(x) \, dx
\end{displaymath}  \pause
$f(x)$ is called the \emph{density function} of $X$. \pause Properties are \pause 
\begin{itemize}
    \item $f(x) \geq 0$ \pause 
    \item $f(x)$ is piecewise continuous. \pause 
    \item $\int_{-\infty}^\infty f(x) \, dx = 1$
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{The probability of any individual value of $X$ is zero}
%\framesubtitle{} 
{\LARGE
\begin{displaymath}
        P(X=a) = 0
\end{displaymath} 
} % End size
\begin{center}
\includegraphics[width=3in]{zero}
\end{center} \pause
\vspace{5mm}

So $P(<X<b) = P(a \leq X<b) = P(a<X \leq b) = P(a \leq X \leq b)$.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
% \frametitle{Since $P(<X<b) = P(a \leq X<b) = P(a<X \leq b) = P(a \leq X \leq b)$} \pause
%\framesubtitle{} 
\begin{center}
    \includegraphics[width=3in]{area}
\end{center}  
{\LARGE
\begin{displaymath}
    P(a<X<b) = F(b)-F(a)
\end{displaymath} 
} % End size
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{$F^\prime(x) = f(x)$} \pause
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3in]{bigF}
\end{center}
\begin{eqnarray*}
    F(x) & = & P(X\leq x) \\
         & = & \int_{-\infty}^x f(t) \, dt 
\end{eqnarray*}\pause

\begin{displaymath}
    \frac{d}{dx} F(x) = \pause \frac{d}{dx} \int_{-\infty}^x f(t) \, dt = \pause
    f(x)
\end{displaymath}  \pause
By the Fundamental Theorem of Calculus. 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{The Fundamental Theorem of Calculus}
%\framesubtitle{} 
$F^\prime(x) = f(x)$ \pause is true for values of $x$ where $F^\prime(x)$ exists \pause
and $f(x)$ is continuous. \pause For example, let \pause
\begin{displaymath}
    f(x) = \left\{ \begin{array}{ll} % ll means left left
   1 & \mbox{for $0 \leq x \leq 1$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.  % Need that crazy invisible right period!
\end{displaymath}  \pause 

\begin{columns}  
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2in]{uniformdensity} \pause
\end{center}
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2in]{uniformCDF} \pause
\end{center}
\end{columns} \vspace{2mm}

$F(x)$ is not differentiable at $x=0$ and $x=1$.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{More comments}
%\framesubtitle{} 
\begin{columns}  
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2in]{uniformdensity} 
\end{center}
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2in]{uniformCDF} \pause
\end{center}
\end{columns} \vspace{2mm}

\begin{itemize}
    \item $F(x)$ is not differentiable at $x=0$ and $x=1$. \pause
    \item These are also the points where $f(x)$ is discontinuous. \pause
    \item The exact value of $f(x)$ at those points cannot be recovered from $F(x)$. \pause
    \item These are events of probability zero. \pause 
    \item They don't really affect anything. \pause
    \item Recall that $f(x)$ is assumed piecewise continuous. \pause
    \item The value of $f(x)$ at a point of discontinuity is essentially arbitrary. \pause This causes no problems.
\end{itemize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{$f(x)$ is not a probability}
\framesubtitle{$g^\prime(x) = \lim_{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}$} \pause
\begin{displaymath}
    f(x) = \lim_{h \rightarrow 0} \frac{F(x+h)-F(x)}{h}
\end{displaymath} 
\begin{center}
\includegraphics[width=2in]{slopeA}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Another way to write $f(x)$}
\framesubtitle{Instead of $\lim_{h \rightarrow 0} \frac{F(x+h)-F(x)}{h}$} \pause
\begin{displaymath}
    f(x) = \lim_{h \rightarrow 0} \frac{F(x+\frac{h}{2})-F(x-\frac{h}{2})}{h}
\end{displaymath} 
\begin{center}
\includegraphics[width=2in]{slopeB}
\end{center} \pause
Limiting slope is the same if it exists.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Interpretation}
%\framesubtitle{} 
{\Large
\begin{displaymath}
    f(x) = \lim_{h \rightarrow 0} \frac{F(x+\frac{h}{2})-F(x-\frac{h}{2})}{h}
\end{displaymath} \pause
\vspace{5mm}

\begin{itemize}
    \item $F(x+\frac{h}{2})-F(x-\frac{h}{2}) = P(x-\frac{h}{2} < X < x+\frac{h}{2})$ \pause
    \item[] 
    \item So $f(x)$ is roughly proportional to the probability that $X$ is in a tiny interval surrounding $x$.
\end{itemize}
} % End size 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Example} \pause
%\framesubtitle{} 

\begin{displaymath}
    f(x) = \left\{ \begin{array}{ll} % ll means left left
   2x & \mbox{for $0 \leq x \leq 1$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.  % Need that crazy invisible right period!
\end{displaymath}  \pause \vspace{5mm}

Common questions: \pause
\begin{itemize}
    \item Prove it's a density.
    \item Find $F(x)$.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Prove it's a density}
%\framesubtitle{} 

\begin{displaymath}
    f(x) = \left\{ \begin{array}{ll} % ll means left left
   2x & \mbox{for $0 \leq x \leq 1$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.  % Need that crazy invisible right period!
\end{displaymath}  \pause \vspace{2mm}

\begin{itemize}
    \item Clearly $f(x) \geq 0$.\pause
    \item It's continuous except at $x=1$.\pause
    \item Show $\int_{-\infty}^\infty f(x) \, dx = 1$
\end{itemize} 
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Show $\int_{-\infty}^\infty f(x) \, dx = 1$}
%\framesubtitle{} 
\begin{displaymath}
    f(x) = \left\{ \begin{array}{ll} % ll means left left
   2x & \mbox{for $0 \leq x \leq 1$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.  % Need that crazy invisible right period!
\end{displaymath}  \pause \vspace{2mm}

\begin{eqnarray*}
    \int_{-\infty}^\infty f(x) \, dx \pause
& = & \int_{-\infty}^0 f(x) \, dx + \int_0^1 f(x) \, dx + \int_1^\infty f(x) \, dx \\ \pause
& = & \int_{-\infty}^0 0 \, dx + \int_0^1 f(x) \, dx + \int_1^\infty 0 \, dx \\ \pause
& = & 0 + \int_0^1 2x \, dx + 0 \\ \pause
& = & \left. 2\frac{x^2}{2} \pause 
\right|_{\raisebox{-1ex}{\scriptsize 0}}^{\raisebox{1ex}{\scriptsize 1}} \\ \pause
% To get that "evaluated at"
& = & 1^2 - 0^2 = 1
\end{eqnarray*} 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Find $F(x) = \int_{-\infty}^x f(t) \, dt$}
%\framesubtitle{} 
\begin{displaymath}
    f(x) = \left\{ \begin{array}{ll} % ll means left left
   2x & \mbox{for $0 \leq x \leq 1$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.  % Need that crazy invisible right period!
\end{displaymath}  \pause \vspace{5mm}

There are 3 cases. \pause
\begin{itemize}
    \item If $x < 0$, \pause $F(x) = \int_{-\infty}^x 0 \, dt \pause =0$. \pause 
    \item If $0 \leq x \leq 1$, \pause 
\begin{displaymath}
    F(x) = \int_{-\infty}^0 0 \, dt + \int_0^x 2t \, dt \pause = x^2. 
\end{displaymath}     \pause
    \item If $x > 1$,  \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$(a,b)$}
\framesubtitle{Parameters $a<b$}
\begin{columns}  
\column{0.5\textwidth}
{\large
\begin{displaymath}
    f(x) = \left\{ \begin{array}{ll} % ll means left left
   \frac{1}{b-a} & \mbox{for $a \leq x \leq b$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.  % Need that crazy invisible right period!
\end{displaymath} \pause
} % End size
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=2.4in]{Uniform}
\end{center}
\end{columns}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{The Exponential Distribution: $X\sim$ Exponential($\lambda$)}
\framesubtitle{Parameter $\lambda > 0$} 
\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 $\lambda > 0$} 
\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($\mu,\sigma$)}
\framesubtitle{Parameters $\mu \in \mathbb{R}$ and $\sigma > 0$} 

\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}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{The Beta Distribution: $X\sim$ Beta($\alpha,\beta$)}
\framesubtitle{Parameters $\alpha>0$ and $\beta > 0$} 
{\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}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
\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(a,b) density with a=0 and b=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)