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\begin{document} 
\title{Assignment 8}
\section{Assignment 8}  
\subsection{Problem 1}
\par\vspace{0.3 cm}
Suppose $X$ and $Y$ are random variables with joint probability density function (joint pdf)
\[
f(x,y)=\frac{x^2+y^2}{26}\quad 0\leq x\leq 2,\quad 0\leq y\leq 3
\]
\par\vspace{0.3 cm}\noindent
a) Show that $f$ is a valid joint density function (i.e., show that $f(x,y)\geq0$ and the integral of $f$ over its support is one)
\par\vspace{0.3 cm}\noindent
b) Find the joint cumulative distribution function (CDF) of $X$ and $Y$
\par\vspace{0.3 cm}\noindent
c) For any function $u(x,y)$,
\[
E(u)=\int_0^2\int_0^3u(x,y)f(x,y)\,dy\,dx
\]
Find the expected values $E(X)$ and $E(Y)$ and the variance-covariance matrix of $X$ and $Y$, defined by:
\[
V_{XY}=\left[\begin{array}{cc}
\sigma_X^2 & \sigma_{XY}\\
\sigma_{XY} & \sigma_Y^2\\
\end{array}\right]
\]
\subsection{Problem 2}
\par\vspace{0.3 cm}
Using the joint density function from problem 1),
\par\vspace{0.3 cm}\noindent
a) Find the marginal PDFs of $X$ and $Y$ and show that they are valid.
\par\vspace{0.3 cm}\noindent
b) Find the marginal CDFs of $X$ and $Y$
\par\vspace{0.3 cm}\noindent
c) Find the expected values $E(X)$ and $E(Y)$ and the variances of $X$ and $Y$ using the marginal PDFs.  How does your answer compare with 1c?
\subsection{Problem 3}
\par\vspace{0.3 cm}
Using the joint density function from problem 1),
\par\vspace{0.3 cm}\noindent
a) Find the conditional PDFs of $X$ given $Y$ and $Y$ given $X$. Show that they are valid.
\par\vspace{0.3 cm}\noindent
b) Find the conditional expected value and variance of $X$ given $Y$ and $Y$ given $X$.
\par\vspace{0.3 cm}\noindent
c) Find conditional expected value of $X$ given that $Y=1$, and the conditional expected value of $Y$ given that $X=1$.\par\vspace{3 cm}\noindent
\subsection{Problem 4}
\vfill
\pagebreak
Suppose $X$ and $Y$ are random variables having a bivariate Bernoulli distribution with the following probabilities:
\par
\begin{tabular}{c|rc}
X: & Y:0 & 1\\
\hline
0   & $p_{00}$ & $p_{01}$\\
1   & $p_{10}$ & $p_{11}$
\end{tabular}
\par\vspace{0.3 cm}\noindent
a) Find the marginal density functions of $X$ and $Y$.
\par\vspace{0.3 cm}\noindent
b) Find the conditional density functions of $X$ given $Y$ and $Y$ given $X$.
\par\vspace{0.3 cm}\noindent
c) Find the expected values of $X$ and $Y$ and their variance-covariance matrix $V_{XY}$.

\end{document}



