Flipping a coin, length of runs

bennyzadir

1. The problem statement, all variables and given/known data

We flip a biased coin (the probability of a head is p, the probability of a tail is q=1-p). Denote X and Y the length of the first and the second run. A "run" is a maximal sequence of consecutive flips that are all the same. For example, if the sequence is HHHTHH... , then X=4, Y=1, if the sequence is THHTHTH..., then X=1, Y=2. Find the followings: E(X), E(Y), E(X^2), E(Y^2), Var(X), Var(Y), Cov(X, Y).

2. Relevant equations



3. The attempt at a solution

I am quite unsure, but I think we should use conditional expeted value conditioning with the result of the first toss. I suppose that to calculate Cov(X, Y) we should use the law of total covariance.

I would be really grateful if you could help me!