Consider the three random variables $X,Y,Z$ in $\mathbb{N}$. Let $Y \sim \operatorname{Ber}(q)$ independent of $X \sim \operatorname{Poi}(\lambda)$.
Determine the probability distribution of $Z=XY$.
How do I solve this question? We have defined the Poisson and Bernoulli distributions in the lecture, however I do not know how to link them in this composition. I am grateful for any hint.
First you have $Y=\text{either } 0\text{ or } 1,$ and if $Y=0$ then $Z=XY=0.$
Thus $\Pr(Z=0) = \Pr(Y=0) + \Pr(X=0\ \&\ Y\ne0) = \Pr(X=0)\cdot\Pr(Y\ne0).$
Then for $z>0$ you have $\Pr(Z=z) = \Pr(Y\ne0\ \&\ X=z)=\Pr(Y\ne0)\cdot\Pr(X=z).$