Let X1, X2, . . . be a sequence of i.i.d. Bernoulli random variables taking value 1 with probability p. Denote by T1,1 the first time the sequence 1, 1 appears:
T1,1 = min{n ∈ N : Xn− 1 = Xn = 1}.
Find E[T1,1] and Var(T1,1)
I solved E[T1,1] by conditioning on X1.
E[T1,1] = E[T1,1 | X1 = 1]P(X1 = 1) + E[T1,1 | X1 = 0]P(X1 = 0)
I know that E[T1,1 | X1 = 1] = E[T1,1 | X1 = 1, X2 = 1]P(X2 = 1) + E[T1,1 | X1 = 1, X2 = 0]P(X2 = 0) = 2(1/2) + (2 + E[T1,1])(1/2)
I also know that E[T1,1 | X1 = 0] = 1 + E[T1,1]
Therefore, E[T1,1] = (1+(2+E[T1,1])(1/2)) + (1+E[T1,1])(1/2) so E[T1,1] = (3/2) + (3/4)(E[T1,1]). Hence, E[T1,1] = 6.
How do I go about solving for Var(T1,1)?