\problemdata                                                             
        {Cold Virus} % latex-friendly label for the prob.
        {Expected Value} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S01 Final, 13} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
A new 24-hour cold is spreading around campus.  On the day they become
infected, a person meets $n$ people, each of whom catch the cold on
the following day independently with probability $p$.  Assume that one
person catches the cold on day 0.

\begin{problemparts}

\problempart Suppose that $r$ people are infected on day $k$. What is the
expected number of people infected on day $k+1$?


\solution{The number of infected people on day $k+1$ is $Y_1 + \cdots
  + Y_r$, where each $Y_i$ is $\mathrm{Bin}(n,p)$. Therefore, the
  expected number of infected people is $nrp$.

  Most people got this one right.}

\problempart Justify and then use Wald's theorem to prove by induction that
the expected number of people infected on day $k$ is $(np)^k$


\solution{  Suppose that $Q_k$ persons have a cold on day~$k$. Let
  $X_{i,k}$ be the number of persons infected by person~$i$ on
  day~$k$. Then
  \begin{displaymath}
    Q_{k+1} = \sum_{i=1}^{Q_k}\sum_{j=1}^n X_{i,j}.
  \end{displaymath}
  Since $Q_k$ and the $X_{i,k}$ are independent, Wald's theorem
  applies, which gives
  \begin{displaymath}
    \Ex(Q_{k+1})
    =
    \Ex\biggl(\sum_{j=1}^n X_{i,j}\biggr) \Ex(Q_k).
    = np \Ex(Q_k).
  \end{displaymath}
  Note that $X_{i,k}$ is $\mathrm{Bin}(n,p)$; therefore $\Ex(X_{i,k})
  = np$ and $\Ex(Q_{k+1}) = np \Ex(Q_k)$. This is a linear recurrence
  for $\Ex(Q_k)$. Given the base case $Q_0 = 1$ it has the solution
  $\Ex(Q_k) = (np)^k$.
  
  Less than 5~people got this one right. Most people argued that since
  the expected number of people is something, these expected people
  are expected to infect some other people, which gives some new
  expected number of infected people. This is not valid reasoning, and
  Wald's theorem is indeed needed to make the argument formal. Others
  thought that the number of infected people is $(np)^k$ on day~$k$,
  when in fact this number is the expectation of the random variable
  denoting the number of infected people on day~$k$.}

\problempart  Suppose that $np < 1$.  Show that the expected total number of
people who catch the cold is finite.  Determine this number.

\solution{  The expected total number of people who catch the cold is
  \begin{displaymath}
    \sum_{k=0}^{\infty} (np)^k = \frac{1}{1 - np}\,.
  \end{displaymath}
  
  Most people got this one, some didn't compute the total number of
  infected, but rather the limit of $(np)^k$ as $k \rightarrow
  \infty$.}

\end{problemparts}
\end{problem}
                                 