Assume that X is uniformly distributed on (0, 1) and that the conditional distribution of Y given X = x is a binomial distribution with parameters (n, x). Then we say that Y has a binomial distribution with fixed size n and random probability parameter.
I have to find EY and Var(Y). I have find a theorem that says that $EY = E(E(Y|X))$ and $VY = V(E(Y|X))+E(V(Y|X))$
But if I have to use this theorem I have first to argue that $E|Y|< \infty$. It seems very obviously, but what will an argument for that be?
If we first try to find EY: First we get that $E(E(Y|X))=E(n \cdot x)$. But how to find this. What is the mean of x? And for VY we get: $VY = V(E(Y|X))+E(V(Y|X))=V(E(n \cdot x))+E(n \cdot x \cdot (1-x))$. But how can I find the mean of these expressions? I hope anyone can help me