% 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: Transformations}}

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 $X \sim$ Poisson($\lambda_1$) and $Y \sim$ Poisson($\lambda_2$) be independent. Using the convolution formula, find the probability mass function of $Z=X+Y$ and identify it by name.
    
\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


    \item Independently for $i = 1, \ldots, n$, let $X_i \sim$ Poisson($\lambda_i$), and let $\displaystyle Y_n = \sum_{i=1}^nX_i$. Using the last problem, what is the probability distribution of $Y_n$?
    
\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item Let $X \sim$ Binomial($n_1,\theta$) and $Y \sim$ Binomial($n_2,\theta$) be independent. Using the convolution formula, find the probability mass function of $Z=X+Y$ and identify it by name.

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


    \item Let $X_1, \ldots, X_n$ be independent Bernoulli random variables with parameter $\theta$, and let $\displaystyle Y_n = \sum_{i=1}^nX_i$. Using the last problem, what is the probability distribution of $Y$?
    
    

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

    \item Let $X$ and $Y$ be independent exponential random variables with parameter $\lambda$. Using the convolution formula, find the probability density function of $Z=X+Y$ and identify it by name. 

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

    \item Let $X_1$ and $X_2$ be independent standard normal random variables. Find the probability density function of $Y_1 = X_1/X_2$.


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

    \item Use the Jacobian method to prove the convolution formula for continuous random variables.

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

    \item Show that the normal probability density function integrates to one.

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

    \item Prove $\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$.  


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

\item \label{Jacobian} The
random variables $X_1$ and $X_2$ are independent. $X_1$ has a gamma distribution with parameters $\alpha=a$ and $\lambda=1$, and $X_2$ has a gamma distribution with parameters $\alpha=b$ and $\lambda=1$. Let $Y_1=\frac{X_1}{X_1+X_2}$ and $Y_2=X_1+X_2$. 
    \begin{enumerate}
% Just leave a full page for solution and determinant.
%        \item Solve for $x_1$ and $x_2$.
%        \item Calculate the determinant.
        \item \label{joint} Give the joint density of $Y_1$ and $Y_2$. Factor, separating $y_1$ and $y_2$ as much as possible. In your final statement of the answer to this part, \emph{specify where the joint density is non-zero.}

\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        
        \item Find $f_{_{Y_1}}(y_1)$, the marginal density of $Y_1$. Again, do not forget to specify where the density is non-zero. \vspace{130mm}
        \item Identify the distribution of $Y_1$ by name; it is on the formula sheet. \vspace{10mm}
      %  \item Are $Y_1$ and $Y_2$ independent? Answer Yes or No and justify your answer in one sentence or less. 
    \end{enumerate}


\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 $F_{xy}(x,y)$. Consider $x<y$ and $x>y$ separately.