## Probability question: random variables

I have two questions.. any insight into either of them is appreciated.

1) A fair coin is tossed repeatedly; the sequence of outcomes is recorded. Let Y be the random time of the first appearance of HT (a tail immediately following a head).
(a) Find P(Y = 4):
(b) Find E[Y] (the expected time to wait for the first HT).
(c) Find the expected length of the interval between successive appearances of HT.
(d) Same as (c) for TT.

2) U is a random variable having mean 2 and variance 5. Two noisy measurements of U are taken:
Y1 = U + Z1
Y2 = U + Z2:
where Z1; Z2; U are assumed pairwise uncorrelated, and where E[Z1] = E[Z2] = 0; Var(Z1) = 1; Var(Z2) = 2:
(a) Determine the linear mean-squared error (MMSE) estimate of U based on Y1 and Y2:
(b) Compute the resulting minimum mean-squared error.

Thank you.
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