Let $X\sim U([0,1])$ and $Y|X ∼ Bin(X, n)$. Find the distribution of $Y$, leaving your answer in terms of the Beta function.

Question

I am trying to solve it, and I am getting that $\mathbb{P}(Y=k)=\frac{1}{n+1}$, so $Y$ is $\mathbb{U}(0,n)$ (discrete uniform distribution).

I am not sure if my final make sense

Answer 1

Note that $Y$ takes value from $\{0,...,n\}$. For any $k \in \{0,...,n\}$ we have:

$$ \mathbb P(Y=k) = \mathbb E[ \mathbb P(Y=k | X)] = \mathbb E[{n \choose k} X^k (1-X)^{n-k}] = {n \choose k} \int_0^1 x^k (1-x)^{n-k}dx $$

The last one is equal to $$ {n \choose k}\mathcal B(k+1,n-k+1) $$ Where $\mathcal B$ is beta function.

It however can be evaluated to ${n \choose k} \frac{\Gamma(k+1)\Gamma(n-k+1)}{\Gamma(n+2)} = {n \choose k} \frac{k!(n-k)!}{(n+1)!} = \frac{1}{n+1}$, so your result is correct, too.