% Sample Question document for STA256
\documentclass[12pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
%\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}
%\pagestyle{empty} % No page numbers


\begin{document}
%\enlargethispage*{1000 pt} 

\begin{center}   
{\Large \textbf{Sample Questions: Conditional Distributions and Independent Random Variables}}

STA256 Fall 2019. Copyright information is at the end of the last page.
%\rule{6in}{.01in} % Width and height
\rule{6in}{.005in} % Horizontal line (Width and height)

% \vspace{3 mm}
\end{center}
\begin{enumerate} 

    \item Let $X$ and $Y$ be continuous random variables. Prove that $X$ and $Y$ are independent if and only if $f_{_{X,Y}}(x,y) = f_{_X}(x) \, f_{_Y}(y)$.
    

\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item Let $X$ and $Y$ be discrete random variables. Prove that if $p_{_{X,Y}}(x,y) = p_{_X}(x) \, p_{_Y}(y)$, then $X$ and $Y$ are independent.


\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item Let $X$ and $Y$ be discrete random variables. Prove that if $X$ and $Y$ are independent, then 
    $p_{_{X,Y}}(x,y) = p_{_X}(x) \, p_{_Y}(y)$. % Need extra paper.


\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item Let {\Large $p_{_{X,Y}}(x,y) = \frac{|x-2y|}{20}$} for $x=1,2,3$ and $y=1,2,3$, and zero otherwise.  \vspace{80mm}
        \begin{enumerate}
            \item What is {\Large$p_{_{Y|X}}(1|2)$}? \vspace{30mm} % 0/5 = 0
            \item What is {\Large$p_{_{X|Y}}(1|2)$}? \vspace{30mm} % 3/6 = 1/2
            \item Are $X$ and $Y$ independent? Answer Yes or No and prove your answer.
        \end{enumerate}

\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item Let 
    {\Large
    $f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll} 
   2e^{-(x+y)} & \mbox{for $ 0 \leq x \leq y$ and $y \geq 0$}  \\
   0     & \mbox{otherwise}  
   \end{array}  \right. $
    } % End size

        \begin{enumerate}
            \item Find $f_{_{X|Y}}(x|y)$. \vspace{120mm}
            \item Are $X$ and $Y$ independent? Answer Yes or No and prove your answer.
        \end{enumerate}

\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item Let $X_1, \ldots, X_n$ be independent random variables with probability density function $f_{_X}(x)$ and cumulative distribution function $F_{_X}(x)$. Let $Y = \max(X_1, \ldots, X_n)$. Find the density $f_{_Y}(y)$. 


\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item Let $X_1, \ldots, X_n$ be independent random variables with probability density function $f_{_X}(x) = e^{-x}$ for $x \geq 0$. Let $Y = \max(X_1, \ldots, X_n)$. Find the density $f_{_Y}(y)$.


\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


    \item Let $X_1, \ldots, X_n$ be independent random variables with probability density function $f_{_X}(x)$ and cumulative distribution function $F_{_X}(x)$. Let $Y = \min(X_1, \ldots, X_n)$. Find the density $f_{_Y}(y)$. 


\end{enumerate}

 \vspace{160mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
\begin{center}\begin{tabular}{l}
\hspace{6in} \\ \hline
\end{tabular}\end{center}
This handout was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational 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: 
     
\begin{center}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
\end{center}


\end{document}

% The answer is a number. Circle your answer.

% MESSY!
    \item Continuing with
    {\Large
    $f_{x,y}(x,y) = \left\{ \begin{array}{ll} 
   2e^{-(x+y)} & \mbox{for $ 0 \leq x \leq y$ and $y \geq 0$}  \\
   0     & \mbox{otherwise}  
   \end{array}  \right. $
   
\noindent
Obtain $F_{_{X,Y}}(x,y)$. Consider $x<y$ and $x>y$ separately.