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\begin{document} 
\title{MTH395 Midterm}
\section{MTH395 Midterm}  
\subsection{Problem 1}
\par\vspace{0.3 cm}
A function $\phi$ is \textit{convex} on the interval $(a,b)$ if for any $x,y\in(a,b)$ and any $\lambda\in(0,1)$,
\[
\phi[\lambda x+(1-\lambda) y]\leq \lambda\phi(x)+(1-\lambda)\phi(y)
\]
If $\phi$ is convex on an interval $(a,b)$ and $Y$ is a random variable that assumes only two values $y_1,y_2\in(a,b)$, neither of which has probability zero, show that
\[
\phi[E(Y)]\leq E[\phi(Y)]
\]
\subsection{Problem 2}
\par\vspace{0.3 cm}
Suppose $(S,\Sigma,\mu)$ is a probability triple associated with an experiment with $S=\{0,1,2,3,\ldots\}$ and $Y$ is a random variable defined on that triple.  Show that
\[
E(Y)=\sum_{y}[1-F(y)]
\]
where $F(y)=P(Y\leq y)$ is the cumulative distribution function of $Y$.
\subsection{Problem 3}
\par\vspace{0.3 cm}
The \textit{correlation} between two random variables $X$ and $Y$ is defined as:
\[
\rho=\frac{\sigma_{XY}}{\sigma_X\sigma_Y}
\]
Show that if $Y_1$ and $Y_2$ are identically distributed (i.e., have the same marginal distribution) then the correlation between $Y_1+Y_2$ and $Y_1-Y_2$ is zero.
\subsection{Problem 4}
\par\vspace{0.3 cm}
An operator $A$ is said to be \textbf{idempotent} if $A(A(x))=A(x)$.  Show that expected value is idempotent in the sense that if $Y$ is a random variable and $E(Y)$ exists, then
\[
E[E(Y)]=E(Y)
\]
\subsection{Problem 5}
\par\vspace{0.3 cm}
If $Y$ is a random vector 
\[
Y=\left[\begin{array}{c}
Y_1\\
Y_2
\end{array}\right]
\]
with joint moment-generating function $M(t_1,t_2)$, the moment-generating function of the random variable $X=Y_1+Y_2$ is $M(t,t)$.  Suppose $Y_1$ and $Y_2$ have joint density function
\[
f(x,y)=e^{-y_1-y_2},\quad 0<y_1,y_2<\infty
\]
Find:
\par\vspace{0.3 cm}\noindent
\begin{tabular}{ll}
a) & The joint moment-generating function $M(t_1,t_2)$ of $Y_1$ and $Y_2$\\
b) & The moment-generating functions of the marginal distributions of $Y_1$ and $Y_2$,\\
& and identify the marginal distributions.\\
c) & The moment-generating function of the sum $Z=Y_1+Y_2$.\\    
\end{tabular}
\subsection{Problem 6}
\par\vspace{0.3 cm}
Show that if $X$ is a discrete random variable and $\mbox{Var}(X)=0$ then there exists an $a\in\mathbb{R}$ such that $P(X=a)=1$
\subsection{Problem 7}
\par\vspace{0.3 cm}
Show that for any random discrete random variables $X$ and $Y$
\[
E[(XY)^2] \leq E(X^2)\cdot E(Y^2)
\]
with equality if and only if $P(aX=bY)=1$ for some $a,b\in\mathbb{R}$ with $a$ and $b$ not both zero. (Hint: consider the random variable $Z=aX-bY$ and use the results of problem 6)
\subsection{Problem 8}
\par\vspace{0.3 cm}
Let $X$ and $Y$ be independently distriuted Poisson random variables with parameters $\lambda$ and $\mu$, respectively.
Show that:
\par\vspace{0.3 cm}
\begin{tabular}{ll}
a) & $Z=X+Y$ has a Poisson distribution with mean $\lambda+\mu$\\ 
b) & The conditional distribution of $X$, given $X+Y=n$ has a binomial distribution\\
c) & Find the parameters of the binomial distribution in part b). 
\end{tabular}
\end{document}



