15. Consider a miner trapped in a room that contains three doors. Door 1 leads him to freedom after two days of travel; door 2 returns him to his room after a four-day journey; and door 3 returns him to his room after a six-day journey. Suppose at all times he is equally likely to choose any of the three doors, and let T denote the time it takes the miner to become free.

(a) Define a sequence of independent and identically distributed random variables X1,X2 . . . and a stopping time N such that

Note: You may have to imagine that the miner continues to randomly choose doors even after he reaches safety.

(b) Use Wald’s equation to find E[T].

(c) Compute E &Ni =1 Xi |N = n '
and note that it is not equal to E[ni =1 Xi].

(d) Use part

(c) for a second derivation of E[T].

Transcribed Image Text:

N T = xi i=1