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\tutorial{tutorial10}{Problems for Tutorial 10}{December 1, 1996}

\section {Wald's Identity and Stopping Rules}

Let $X_1,X_2\ldots$ be a sequence of iid random variables. A 
{\em stopping rule} for a set of random variables $\{X_n;n \geq 1 \}$
is a random variable $N$ with values in $\naturals$,
such that the event $\{N \geq n \}$ may depend on $X_1,\ldots,X_{n-1}$ but
is independent of the values $X_{n},X_{n+1},\ldots$. \\
Prove Wald's Identity for stopping rules: \\
If $X_1,X_2,\dots$ and $N$ are as above, then 

\begin{displaymath}
E[ \sum_{i=1}^{N}{X_i}] = E[X]E[N]
\end{displaymath}

{\it (Hint: Let $I_n$ be the indicator of the event $\{N \geq n \}$ and 
consider the sum $\sum_{n=1}^{\infty}{X_nI_n}$ )} 
   
\section{The Distressed Student}

A Harvard student, lost in the underground corridors of MIT, finds himself in a room 
containing three doors.  Door 1 leads him to freedom (out of MIT) after two 
hours travel; door 2 returns him to the his room after four hours travel, and 
door 3 returns him to the room after eight hours travel.  Each time he finds himself in the room, the Harvard student picks a door with equal probability.  
Let $T$ denote the time until he finds the exit. \\

\bparts

\ppart Define a sequence of iid random variables $X_i$ and a stopping rule 
$N$ such that

\begin{displaymath}
T = \sum_{i=1}^{N}{X_i}
\end{displaymath}

\ppart Use Wald's Identity to find $E[T]$

\ppart Now suppose that an MIT student is also lost and finds herself in a 
similar situation to the Harvard student.  The MIT student never choses the 
same door twice.  What is $E[T]$ for the MIT student?

\eparts   

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