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.