Marginal of Conditional Binomial?

[Xkhaos]

Let X~Bin(n,p), Y|X=x~Bin(x,p*)
Is there a shortcut to get P(Y=y)?
I ask because I'm having trouble getting P(Y=y) through the use of
P(Y=y) = sum of all x of f(x,y), where f(x,y) = f(y|x)*f(x).

[Academic Actuary]

This is a guess but it would seem that y would be binomial (n, pp*).

[daaaave]

Suppose you flip n coins, each of which is heads with probability p. Then X is the number of heads. Saying that (Y | X=x)~Bin(x, p*) means that we then flip X new coins, each of which is heads with probability p*, and Y is the number of heads on that second round. Then Y~Bin(n, pp*) because each original flip results in heads on the second round with probability pp*.

[asrauf]

Originally Posted by daaaave

Suppose you flip n coins, each of which is heads with probability p. Then X is the number of heads. Saying that (Y | X=x)~Bin(x, p*) means that we then flip X new coins, each of which is heads with probability p*, and Y is the number of heads on that second round. Then Y~Bin(n, pp*) because each original flip results in heads on the second round with probability pp*.

Wonderful explanation Dave!
I passed P in May mainly through your well explained concepts...