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\begin{document}

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

Directions: {\bf \Large Work only on this sheet} (on both sides, if
needed); do not turn in any supplementary sheets of paper. There is
actually plenty of room for your answers, as long as you organize
yourself BEFORE starting writing.

{\bf \Large Unless otherwise stated, give numerical answers as
expressions, e.g. $\frac{2}{3} \times 6 - 1.8$.  Do NOT use
calculators.}

{\bf 1.} (15) Consider random variables $X_1$ and $X_2$, for which
$Var(X_i) = 1.0$ for i = 1,2, and $Cov(X_1,X_2) = 0.5$.  Find
$Var(X_1+X_2)$.

{\bf 2.} (15) Suppose we have random variables X and Y, and define the new
random variable Z = 8Y.  Then which of the following is correct?
(i) $\rho(X,Z) = \rho(X,Y)$.  (ii) $\rho(X,Z) =
0$. (iii) $\rho(Y,Z) = 0$.  (iv) $\rho(X,Z) = 8 \rho(X,Y)$.  (v)
$\rho(X,Z) = \frac{1}{8} \rho(X,Y)$. (vi) There is no special
relationship.

{\bf 3.} Suppose $f_X(t) = 2t$ for $0 < t <1$ and the density is 0
elsewhere.

\begin{itemize}

\item [(a)] (10) Find $h_X(0.5)$.

\item [(b)] (10) Which statement concerning this distribution is correct?
(i) IFR. (ii) DFR. (iii) U-shaped failure rate. (iv) Sinusoidal failure
rate. (v) Failure rate is undefined for $t > 0.5$.

\end{itemize}

{\bf 4.} (15) Consider the coin game on p.33.  Find $F_{X_3,Y_3}(0,0)$.

{\bf 5.} (15) In the backup battery example on p.85, find Var(W).

{\bf 6.} (10) Consider the ``8st'' density example on p.74.  Find $P(Y >
X^2)$.  Express your answers as a definite integral, ready for any
calculus student to compute an actual number from.

{\bf 7.} (10) What will be the (approximate) output of the following R
code?

\begin{Verbatim}[fontsize=\relsize{-2}]
s <- 0
s2 <- 0
for (rep in 1:10000) {
   z3 <- rnorm(3)  # generate 3 N(0,1) random variates
   tot <- sum(z3^2)  # sum of the squares of the 3 variates
   s <- s + tot
   s2 <- s2 + tot^2
}
m <- s/10000
print(m)
print(s2/10000 - m^2)
\end{Verbatim}

{\bf Solutions:}

{\bf 1.} 3

{\bf 2.} (i)

{\bf 3a.} $h_X(t) = 2t/(1-t^2)$, so $h_X(0.5) = 4/3$

{\bf 3b.} IFR

{\bf 4} $F_{X_3,Y_3}(0,0) = P(X_3 \leq 0 \textrm{ and } Y_3 \leq 0)
= P(X_3 = 0 \textrm{ and } Y_3 = 0) = 0.5^3 \cdot 0.5$

{\bf 5.} $Var(W) = Var(X+Y) = Var(X) + Var(Y) = 2^2 + 1^2$

{\bf 6.} 
$$
\int_{0}^{1} \int_{s^2}^{s} 8st ~ dt ~ ds
$$

{\bf 7.} Using Sections 2.3.3.1 and 2.3.5.1, and (1.4.6), we
have that the output will be 3 and 6.
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