A fair coin is tossed n + 1 times. For 1 ≤ i ≤ n, let $A_i$ be 1 if the ith and (i + 1)st outcomes are both heads, and 0 otherwise.
a) Find the mean and the variance of $A_i$.
b) Find the covariance of Ai and Aj for i != j.
c) Define $M = A_1+· · ·+A_n$, the number of occurrences of the motif HH in the sequence. Find the mean and variance of M.
d) Use a similar method to find the mean and variance of the number of occurrences of the motif TH in the sequence.
Can anyone help me with this question? Here's what I have so far:
a) Mean: $\frac{1}{4}$, variance: $\frac{3}{16}$
b) $\text{Cov}(A_i, A_{i+1}) = \frac{1}{16}$
c) $E(M) = nE(A_i) = n/4$, $Var(M) = nVar(A_1) + 2nCov(A_i, A_{i+1}) = \frac{3n}{16}+\frac{2n}{16}=\frac{5n}{16}$
d) Repeating the same for TH, I get $E(M) = n/4$, $Var(M) = \frac{3n}{16}-\frac{2n}{16}=\frac{n}{16}$.
I'm not sure if any of this is on the right track so any comments/suggestions would be appreciated :)