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\begin{document} 
\title{Assignment 7}
\section{Assignment 7}  
\subsection{Problem 1}
\par\vspace{0.3 cm}
If $a$ and $b$ are constants and $Y$ is a discrete random variable with moment-generating function $m_Y(t)$, show that the moment-generating function of $X=aY+b$ is
\[
m_X(t)=e^{tb}m_Y(at)
\]
\subsection{Problem 2}
\par\vspace{0.3 cm}
Suppose $Y$ is a random variable with moment-generating function $m(t)$ that is finite for $|t|\leq b$.  Show that
\[
P(X\geq a) \leq e^{-at}m(t)\quad\mbox{for}\quad |t|\leq b
\]
(Hint: use Markov's inequality)
\subsection{Problem 3}
\par\vspace{0.3 cm}
Let $Y$ be a random variable having a Poisson distribution with parameter $\lambda$.  
\par\vspace{0.3 cm}
a) Find $E[Y(Y-1)]$
\par\vspace{0.3 cm}
b) Use the results of part a) to show that $V(Y)=\lambda$.
\subsection{Problem 4}
\par\vspace{0.3 cm}
Suppose $Y_n$ is a sequence of random variables such that $Y_i$ has a binomial $B(i,p)$ distributon for some $p\in[0,1]$.  Show that for any positive constant $c\in(0,1)$,
\[
P(|Y_i-ip|\geq cn)\rightarrow0\quad\mbox{as}\quad n\rightarrow\infty
\]

\end{document}



