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\begin{document} 
\title{Assignment 2}
\section{assignment 2}
\subsection{Problem 1}
Suppose $Y=(Y_1,\ldots,Y_n)$ is a vector of IID $N(0,1)$ random variables.
\par\vspace{0.3 cm}\noindent
a) Find the distribution of
\[
\sum_{i=1}^nY_i^2
\]
\par\vspace{0.3 cm}\noindent
b) Define
\[
W_n=\frac{1}{n}\sum_{i=1}^nY_i^2
\]
Show that $W_n$ converges in probability to a constant, and find the value of the constant.
\subsection{Problem 2}
\par\vspace{0.3 cm}
Suppose $Y=(Y_1,\ldots,Y_n)$ is a vector of IID $N(\mu,1)$ random variables.
\par\vspace{0.3 cm}\noindent
a) Show that $Y_1$ is an unbiased estimator for $\mu$
\par\vspace{0.3 cm}\noindent
b) Find
\[
P(|Y_1-\mu|\leq1)
\]
\par\vspace{0.3 cm}\noindent
c) Based on the result of b), is $Y_1$ a consistent estimator for $\mu$?
\subsection{Problem 3}
\par\vspace{0.3 cm}
Suppose $Y$ is a random variable with a binomial distribution having $n$ trials and probability of success $p$.  Show that  
\[
\frac{1}{n}Y
\]
is a consistent estimator for $p$.
\subsection{Problem 4}
\par\vspace{0.3 cm}
Let $Y=(Y_1,\ldots,Y_n)$ be a random sample of size $n$ where each $Y_i$ has the density function
\[
f(y)=\left\{\begin{array}{lcl}
\theta y^{\theta-1} &\mbox{if}& 0<y<1\\
0 &\mbox{otherwise}&\\
\end{array}\right.
\]
where $\theta>0$.  Show that 
\[
\overline{Y}=\frac{1}{n}\sum_{i=1}^nY_i\quad\mbox{is a consistent estimator of}\quad\frac{\theta}{\theta+1}
\]
\end{document}



