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\title{Tutorial 10 Solutions}

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\section{Wald's Identity}

Let $I_n$ be the indicator of $\{ N \geq n \}$. Let 

\begin{displaymath}
S_N = \sum_{n=1}^{N}{X_x}
\end{displaymath}

Note that since $I_n = 1$ if $N \geq n$ and $I_n = 0$ if $N \leq n-1$ 
the above sum can be written as

\begin{displaymath}
S_N = \sum_{n=1}^{\infty}{X_nI_n}
\end{displaymath}

Then we have

\begin{displaymath}
E[S_N] = \sum_{n=1}^{\infty}{E[X_nI_n]} =
E[X] \sum_{n=1}^{\infty}{E[I_n]} =
E[X] \sum_{n=1}^{\infty}{P( N \geq n)} =
E[X]E[N]
\end{displaymath}

\section{Problem 2}

\bparts

\ppart Let $X_i$ take the values $2,4,8$ with probability $1/3$, and let
$N= \min \{n | X_n=2 \}$.  $N$ is geometric with mean $3$

\ppart $E[T] = E[N]E[X] = 3(14/3) = 14$

\ppart $E[T] = (1/3)(2) + (1/3)(4+(1/2)(2+10)) + (1/3)(8+(1/2)(2+6)) = 8$

\eparts


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