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\newtheorem*{example}{Example}
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\begin{document} 
\title{Assignment 1}
\section{assignment 1}
\subsection{Problem 1}
Suppose $Y$ is a random variable with support on $[0,\infty)$ and density function $f_y(y)$.  Define an new random variable
\[  
Z=g(Y)\quad\mbox{where}\quad X=g(Y)=kY,\quad k>0
\]
\par\vspace{0.3 cm}\noindent
a) Find the density function of $Z$, $f_z(z)$.
\par\vspace{0.3 cm}\noindent
b) Use the density functions of $Y$ and $Z$ to show that for $b>0$,
\[
P(0\leq Y \leq b)\;=\;P(0\leq Z \leq kb)
\]
\subsection{Problem 2}
\par\vspace{0.3 cm}
Suppose $Y=(Y_1,\ldots,Y_n)$ is a vector of IID exponential random variables with parameter $\theta$, and let $S=\sum_{i=1}^nY_i$.
\par\vspace{0.3 cm}\noindent
a) Find the density function of $S$ (hint: use moment-generating functions) 
\par\vspace{0.3 cm}\noindent
b) Find the expected value and MSE of
\[
\hat{\theta}=\frac{S}{n}
\]
\subsection{Problem 3}
\par\vspace{0.3 cm}
Suppose $Y=(Y_1,\ldots,Y_n)$ is a vector of IID exponential random variables with parameter $\theta$, and let
\[
Y_{(1)}=\min(Y_1,\ldots,Y_n)
\]
\par\vspace{0.3 cm}\noindent
a) Find the density function of $Y_{(1)}$
\par\vspace{0.3 cm}\noindent
b) Find expected value and MSE of 
\[
\hat{\theta}\;=\;n\cdot {Y_(1)}
\]
\subsection{Problem 4}
\par\vspace{0.3 cm}
Suppose $\hat{\theta_1}$ and $\hat{\theta_2}$ are estimators of $\theta$ with
\[
E(\hat{\theta_1})=E(\hat{\theta_2})=\theta,\quad V(\hat{\theta_1})=\sigma_1^2,\quad\mbox{and}\;V(\hat{\theta_2})=\sigma_2^2
\]
Find the value of the constant $a$ that minimizes
\[
V(\hat{\theta_3})\quad\mbox{where}\quad\hat{\theta_3}\;=\;a\cdot\hat{\theta_1}+(1-a)\cdot\hat{\theta_2}
\]

\end{document}



