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@André: In your shorter solution, are you using a property like: if $X = f(Y)$, then $P(X = f(y)) = P(Y = y)$? If yes, why is it true? Is it true only for one-to-one functions?
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@André: Separating the even and odd case was the trick here, I guess. I was completely stuck thinking how to write a common expression for the PMF. Thanks!
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Edited the question. Thanks for the detailed solution. Any hints on how to generalize if $P(H) = p$?
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