probability

Exercise

A fair die is rolled repeatedly .Let Xn be the result of the nth roll. So Xn takes values {1, . . . , 6}, each with probability 1/6, and the random variables X1, X2, . . . are all independent.

That is, N is the first roll where the result is equal to the previous roll. [e.g., if you roll the sequence 2,3,1,4,4,6,... then N = 5.] Find E[N].

N=min{n:Xn =Xn−1,n≥2}

[Hint: Hint: consider the expected value conditioned on the first roll being equal to k. By conditioning on the second roll, get a recursion relation to compute this expected value. Then undo the conditioning on the first roll.]