# Using Wald's Equation.

I am trying to solve the following problem.

An underground explorer lost in a cave is faced with three potential exit routes. Route 1 will take him to freedom in 2 hours; route 2 will return him to the cave in 4 hours; and route 3 will return him to the cave in 6 hours. Suppose at all times he is equally likely to choose any of the three exits, and let $T$ be the time it takes the explorer to reach freedom. Define a sequence of iid variables $X_1,X_2,\cdots$ and a stopping time $N$ such that $T=\sum _{i=1}^{N}X_i$ and use Wald's Equation to find $E(T)$.

Solution Attempt $N$ is a stopping time with $E(N)<\infty$ and $E(X) = 2.\dfrac {1} {3}+4.\dfrac {1} {3}+6\cdot \dfrac {1} {3}=4 < \infty$. Applying Wald's Equation gives $E(T = \sum _{i=1}^{N}X_i)=E(X)E(N) = 4E(N)$

How can we compute $E(N)$ ?

Thanks.

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Your solution attempt fails to follow the instructions. You're asked first to "define a sequence of iid variables $X_1, X_2, \ldots$", but there is no such definition in your solution. (Here "define" implies that you must explain what the interpretation of each of those variables is in terms of the explorer and the caves). – Henning Makholm Oct 24 '12 at 20:12

$$E(N)=\sum_{n=1}^{\infty} P(n \text{ routes before exit})\cdot n$$
$$E(N)=\sum_{n=1}^{\infty} \frac{1}{3}\cdot\left(\frac{2}{3}\right)^{n-1}\cdot n$$
$$E(N)=3$$