I'm struggling with understanding why the following statement is true:
Let $X$ be a Random Variable with Poisson distribution. Let $Z$ be a Random Variable independent from $X$, whose distribution is $P(Z=0.9)=0.2=1-P(Z=0.6)$. Let $Y$ be a Random Variable such that $Y |( X=x , Z=z)\sim \text{Binom}(x,z)$.
Then given $Z=0.9, Y\sim \text{Poisson}$.
I'm told that it is a consequence of some general fact about split Poisson variables but I couldn't make much of that fact, or wasn't able to see why it's true by myself. I don't really know how to go about this so any help would be greatly appreciated.
Thanks!