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\begin{document}

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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.} (10) Suppose $X$ is the length of a random rod, in inches, and
$Var(X) = 2.6$.  Let $Y$ denote the length in feet.  Find $Var(Y)$.

{\bf 2.} (10) In the board game, Sec. 2.11, suppose we start at square 3
(no bonus, since we {\it start} there rather than {\it landing} there).
Let $X$ denote the square we land on after one turn.  Find $EX$.

{\bf 3.} This problem concerns the Monty Hall example, pp.40ff.

\begin{itemize}

\item [(a)] (15) Give the numbers of the ``mailing tubes'' in (3.1) and
(3.2), respectively.  Use a comma and/or spaces to separate the two
equation numbers, e.g.  ``(2.1) (2.3)''.

\item [(b)] (15) Consider (3.1).  Say we change the left-hand side to $P(A
= 2 ~|~ C = 2, H = 1)$.  What would be the new numerical value of the
numerator on the right-hand side?

\end{itemize}

{\bf 4.} (20) Look at the simulation code on p.26.  Say we wish to find
the expected value of $S^2$, where $S$ is the sum of the {\bf d} dice.
Give a line of code, to replace line 11.

{\bf 5.} Consider the Preferential Attachment Graph model, Sec. 2.13.1..

\begin{itemize}

\item [(a)] (10) Give the number of the ``mailing tube'' justifying (2.69).

\item [(b)] (10) Find $P(N_3 = 1 ~|~ N_4 = 1)$.

\item [(c)] (10) Find $P(N_4 = 3)$.

% \begin{equation}
% \left (
% \begin{array}{rrr}
% 0 & 1 & 0 \\
% 1 & 0 & 1 \\
% 0 & 1 & 0 \\
% \end{array}
% \right )
% \end{equation}

% Find the new value of (2.70).`

\end{itemize}

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\newpage

{\bf Solutions:}

{\bf 1.}

$$
(\frac{1}{12})^2 \cdot 2.6
$$

{\bf 2.}

$$
4 \cdot \frac{1}{6} +
5 \cdot \frac{1}{6} +
6 \cdot \frac{1}{6} +
7 \cdot \frac{1}{6} +
0 \cdot \frac{1}{6} +
1 \cdot \frac{1}{6} 
$$

{\bf 3.a} (2.8), (2.7)

{\bf 3.b}

$$
( \frac{1}{3})
( \frac{1}{3})
(\frac{1}{2})
$$

{\bf 4.}

\begin{lstlisting}
mean(sums^2)
\end{lstlisting}

{\bf 5.a} (2.2)

{\bf 5.b} 

$$
(1/2) (2/4) ~/~ ((1/2) (2/4) + (1/2) (1/4))
$$

{\bf 5.c} 

$$
(1/2) (1/4) + (1/2) (1/4)
$$

\end{document}