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{\bf\underline{MATH 323. TEST III}}
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{\bf NAME:}

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Directions: You may use your own calculator and your own textbook. You may also use a summary (one side of an 8.5"x11" sheet of paper 
with notes in your writing). You may use nothing else. You may not pass a calculator, textbook or summary to another person. 
To receive full credit you must show all your work. Erase or cross out any work you do not want graded. 


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1.(20 points) Let $X$ and $Y$ be random variables with joint density function 
\[ f(x,y)=\left\{ \begin{array}{ll}
                    \frac{3}{4}\,y      & \mbox{, if $x\geq0\;$, $y\geq0\;$, $0\leq x+y\leq2\;,$} \\
                    0 & \mbox{, elsewhere.}               
                 \end{array}
         \right. \]
Are $X$ and $Y$ independent? Prove your assertion.

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2.(10 points) Let $X$ and $Y$ be independent random variables with moment generating functions $M_X(t)=1+t$ and 
$M_Y(t)=e^t$. Find the moment generating function for $U=2X+Y$, and use it to compute $E(U)$ .

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3.(20 points) Let $X$ and $Y$ be random variables with joint density function 
\[ f(x,y)=\left\{ \begin{array}{ll}
                    1 & \mbox{, if $0\leq x\leq2\;$, $0\leq y\leq1\;$, $2y\leq x$} \\
                    0 & \mbox{, elsewhere}               
                 \end{array}
         \right. \]
a) Find the marginal density $f_X(x)$ of $X$.

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b) Find $P(-1\leq Y\leq1/4\,|\,X=1)$ .

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c) Compute $E(Y\,|\,X=x)$ as a function of $x$ .

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4.(20 points) Assume $X$ and $Y$ are jointly uniformly distributed over the unit square $[0,1]\times[0,1]$ and 
let $U=Y/X$. Find the density function $f_U(u)$ of $U$. 

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5.(15 points) Let $X_1$ and $X_2$ be random variables such that $E(X_1)=E(X_2)=2$, $E(X_1^2)=E(X_2^2)=7$, and 
$E(X_1X_2)=3$. Find $V(X_1-2X_2)$ .

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6.(15 points) Suppose ten people each toss two coins. Let $Y_1$ denote the number of people who got two tails, 
$Y_2$ the number of people who got two heads, and $Y_3$ the number of people who got a head and a tail.\\
a) Find $P(Y_1=2,\;Y_2=3,\;Y_3=5)$ .\\
b) Find $V(2Y_1+5Y_3)$ .  


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