\documentclass[12pt]{amsart}
\usepackage{graphicx}
\theoremstyle{plain}    
\newtheorem*{definition}{Definition}
\newtheorem*{example}{Example}
\newtheorem*{exercise}{Exercise}
\newtheorem*{conjecture}{Conjecture}
\newtheorem*{theorem}{Theorem}
\newtheorem*{lemma}{Lemma}
\newtheorem*{remark}{Remark}
\begin{document} 
\title{Assignment 9}
\section{Assignment 9}  
\subsection{Problem 1}
\par\vspace{0.3 cm}
Suppose $X$ random variables with density function (PDF)
\[
f(x)=1\quad 0<x<1
\]
\par\vspace{0.3 cm}\noindent
a) Find the density function of the random variable $Y$ defined by 
\[
Y=X^2
\]
\par\vspace{0.3 cm}\noindent
b) Find the cumulative distribution function (CDF) of $Y$
\par\vspace{0.3 cm}\noindent
c) Find $E(Y)$
d) Find $V(Y)$
\subsection{Problem 2}
\par\vspace{0.3 cm}
Suppose $Y$ has the \textit{double exponential} distribution
\[
f(y)=\frac{1}{2}e^{-|y|}\quad -\infty<y<\infty
\]
\par\vspace{0.3 cm}\noindent
a) Find the density function (PDF) of the random variable $X=|Y|$
\par\vspace{0.3 cm}\noindent
b) Find the cumulative distribution function (CDF) of $X$ 
\par\vspace{0.3 cm}\noindent
c) Find the expected value $E(X)$
\par\vspace{0.3 cm}\noindent
d) Find the variance $V(X)$
\subsection{Problem 3}
\par\vspace{0.3 cm}
Let $Z_1$ and $Z_2$ be independent random variables each having the standard normal (i.e., $N(0,1)$) distribution. 
\par\vspace{0.3 cm}\noindent
Find the joint density function of the random variables
\[
U_1=Z_1\quad\mbox{and}\quad U_2=Z_1+Z_2
\]
\par\vspace{0.3 cm}\noindent
\subsection{Problem 4}
Suppose $Y_1$ and $Y_2$ are independent exponentially distributed random variables with common mean $\beta$. Show 
that the joint density of $U_1=Y_1+Y_2$ and $U_2=Y_1/Y_2$ is
\[
f_{U_1,U_2}(u_1,u_2)=\left\{\begin{array}{lcl}
\frac{1}{\beta}u_1e^{-u_1/\beta}\frac{1}{(1+u_2)^2} & 0<u_1,\;0<u_2\\
0 & \mbox{otherwise}\\
\end{array}\right.
\]

\end{document}



