Probability theory: If $X|Y=p $ is a first success distribution with parameter $p$ and $Y \in U(0,1)$. What's the distribution of X?

Question

I've tried using approximations for expected values of functions and varaince of function. But it doesn't seem to add up to something I recognize.

I think I'm doing something wrong. Should I maybe not use approximations here? Thanks in advance!

Answer 1

Hint:$$P\left(X=k\right)=\int P\left(X=k\mid Y=y\right)f_{Y}\left(y\right)\;dy=\int_{0}^{1}P\left(X=k\mid Y=y\right)\;dy=\cdots$$