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\mbox{{\sc Mathematics 445/545 - Applied Probability - Fall 2000 - Zirbel}}\\
{\bf Homework \#4. Due Friday, October 6.}
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\blist{0.2in}

\item Suppose $X_{1}, X_{2}, \ldots$ all have the same mean and take values
in $[0,\infty)$.
Let $N$ take values in the set $\{0, 1, 2, \ldots\}$ and suppose that $N$
is independent of $X_{i}$ for $i=1, 2, 3, \ldots.$
Define a new random variable $S$ by $S = X_{1} + \cdots + X_{N}$; when it
happens that $N=0$, then we set $S=0$.

The steps below show that, in this case, $\E [S] = \E [N] \E [X_{1}]$.
Your job is to rewrite these steps and {\em provide justifications for each
step}.

The notation $\ind$ means a random variable whose value is 1 when $N=n$
and 0 otherwise.
\bda
    \E [S] 
    &=& \E \left[ \left( \nsum \ind \right) S \right] \\
    &=& \nsum \E \left[ \ind S \right] \quad \mbox{because $\ind S \geq 0$}\\
    &=& \nsum \E \left[ \ind (X_{1} + \cdots + X_{N}) \right] \\
    &=& \nsum \E \left[ \ind (X_{1} + \cdots + X_{n}) \right] \\
    &=& \nsum \isum \E \left[ \ind \Xi \right] \\
    &=& \nsum \isum \E \left[ \ind \right] \E \left[ \Xi \right] \\
    &=& \nsum \isum \P(N=n) \E \left[X_{1} \right] \\
    &=& \nsum n \P(N=n) \E \left[X_{1}\right] \\
    &=& \E[N] \E[X_{1}]
\eda
This result was used in computing the expected waiting time in our discrete
time queueing model.

\item For the discrete time queueing model we have been discussing in
class, we found that $W = 1/(p-\lambda)$.
Write a Matlab program called {\tt wait.m} which produces a graph of this
relationship.
Add comments to each line of the program to explain what is being done.
Print out the graph and the program.

Some suggested commands to use and how to get help on them:
\begin{tabbing}
\qquad \= {\tt lambda = 0:0.01:p;}   \qquad \= help colon\\
\>        {\tt W = 1./(p-lambda);}          \>help slash\\
\>        {\tt plot(lambda,W);}             \>help plot\\
\>        {\tt hold on}                     \>help hold
\end{tabbing}
You should change the horizontal axis to display values out to $\lambda =
1$ using {\tt axis}.
Add a vertical dashed line at $p$ using a command similar to {\tt
plot([p p], [0 10], '-');}.
Add labels to the axes and a title for the graph.

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