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Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Directions: MAKE SURE TO COPY YOUR ANSWERS TO A SEPARATE SHEET FOR SENDING
ME AN ELECTRONIC COPY LATER.

{\bf 1.}  Consider Equation (3.64).

\begin{itemize}

\item [(a)] (15)
List (on one line), the equation number(s) of the mailing tubes
used to justify the equality $Var(7+2I) = 4 Var(I)$. 

\item [(b)] (15) Give the equation number of the relation that justifies
$4 Var(I) = 4 \cdot 0.5 (1-0.5)$.

\end{itemize}

{\bf 2.}  (15) Give the number of the mailing tube that justifies (3.80).

{\bf 3.}  Consider the variables $G_i$, p.56.

\begin{itemize}

\item [(a)] (10) Find $P(G_2 = 1 ~|~ G_1 = 1)$.

\item [(b)] (15) Find $P(G_1 = G_2)$.

\end{itemize}

{\bf 4.} (15) Suppose $X$ and $Y$ are independent random variables, with
$EX = 1$, $EY = 2$, $Var(X) = 3$ and $Var(Y) = 4$.  Find $Var(XY)$.

{\bf 5.}  (15) In a certain game, Person A spins a spinner and wins $S$ dollars,
with mean 10 and variance 5.  Person B flips a coin. If it comes up
heads, Person A must give B whatever A won, but if it comes up tails, B
wins nothing.  Let $T$ denote the amount B wins.  Find $Var(T)$.

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{\bf Solutions:}

{\bf 1.a} (3.47), (3.40)

{\bf 2.} (3.32)

{\bf 3.a} Given the first draw resulted in a man, there will be 5 men
and 3 women left, so the probability is 5/8.

{\bf 3.b} The requested probability is that of getting two men or two
women, (6/9) (5/8) + (3/9)(3/8)

{\bf 4.}  Use the relations $E(UV) = EU \cdot EV$ (for independent U,V)
and then use $Var(U) = E(U^2) - (EU)^2$ repeatedly:

\begin{eqnarray}
Var(XY) &=& E(X^2 Y^2) - [E(XY)]^2 \\
&=& E(X^2) \cdot E(Y^2) - (EX \cdot EY)^2 \\ 
&=& [Var(X) + (EX)^2] \cdot
[Var(Y) + (EY)^2] - (EX \cdot EY)^2 \label{genform} \\
&=& (3 + 1^2) (4 + 2^2) - (1 \cdot 2)^2
\end{eqnarray}

{\bf 5.} Use (\ref{genform}), in this case with X = I, where I is an
indicator variable for the event that B gets a head, and Y = S.  Then T
= IS, and I and S are independent, so

\begin{equation}
Var(T) = Var(IS) = 
[Var(I) + (EI)^2] \cdot
[Var(S) + (ES)^2] - (EI \cdot ES)^2 
\end{equation}

Then use the facts that I has mean 0.5 and variance 0.5(1-0.5), with S
having the mean and variance given in the problem.

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