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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.} (20) Fill in the blank:  Density functions for continuous random
variables are analogs of the
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
functions that are used for discrete random variables.

{\bf 2.} (20) Suppose for some random variable W, $F_W(t) = t^3$ for $0 <
t < 1$, with $F_W(t)$ be 0 and 1 for $t < 0$ and $t > 0$, respectively.
Find $f_W(t)$ for $0 < t < 1$.

{\bf 3.} (10) Suppose X has a binomial distribution with parameters n and
p.  Then X is approximately normally distributed with mean np and
variance np(1-p).  For each of the following, answer either A or E, for
``approximately'' or ``exact,'' respectively:  

\begin{itemize}

\item [(a)] (10) distribution of X is normal

\item [(b)] (10) E(X) is np

\item [(c)] (5) Var(X) is np(1-p)

\end{itemize}

{\bf 4.} Suppose light bulb lifetimes have an exponential distribution
with mean 100.0 hours, i.e. $\lambda = 0.01$.  We use our first light
bulb, with it lasting for $X_1$ hours.  When it burns out, we replace it
with a second bulb, which lasts $X_2$ hours.  Then $T_2 = X_1 + X_2$ is
the time of the second replacement.  

\begin{itemize}

\item [(a)] (10) Give numerical expressions for the mean and variance of
$T_2$.

\item [(b)] (5) State $f_{T_2}(t)$ (the actual function, not the name of
a family etc.).

\item [(c)] (10) Fill in the blank:  

$$
P(T_2 > t) = P[N(t) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ]
$$

\end{itemize}

{\bf Solutions:}

{\bf 1.} probability mass functions

{\bf 2.} $3t^2$

{\bf 3a.} A (pages 30-31)

{\bf 3b.} E (pages 30-31)

{\bf 3c.} E (pages 30-31)

{\bf 4a.} $2 \cdot 100$, $2 \cdot 100^2$ (page 59)

{\bf 4b.} $0.01^2 t e^{-\lambda t}$ (page 59)

{\bf 4c.} $\leq 1$ (like (2.38))

\end{document}