I toss 3 fair coins, and then re-toss all the ones that come up tails. Let X denote the number of coins that come up heads on the first toss, and let Y denote the number of re-tossed coins that come up heads on the second toss. (Hence 0 ≤ X ≤ 3 and 0 ≤ Y ≤ 3 − X.)
(a) Determine the joint pmf of X and Y , and use it to calculate E(X + Y ).
(b) Derive a formula for E(Y |X) and use it to compute E(X + Y ) as E(E(X + Y |X)).
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