Find the probability generating function of $Z=\sum_{i=1}^{N}X_i.$

Question

Let $X_1,X_2,...$ be a sequence of i.i.d Bernoulli random variables with parameter $p$. Let $N$ be a Poisson random variable with with parameter $\lambda$ which is independet of the $X_i.$

(a) Find the probability generating function of $Z=\sum_{i=1}^{N}X_i.$

(b) Use (a) to identify the distribution of $Z$.

The pgf of a sum of i.i.d random variables is the product of their respective pgf's so:

$$G_Z(s)=G_{X_1}(s)\cdot G_{X_2}(s)\cdot...\cdot G_{X_N}(s),$$

and sice the $X_{i}$ are Bernoulli distributed with parameter $p$ we have that

$$G_Z(s)=(1-p+ps)^N.$$

I'm not sure how to proceed from here. I can't express my $G_Z(s)$ in terms of a random variable, so what should I do with the $N$?

Answer 1

Long story short, it ends up being the expectation of what you have written. But you should start from scratch and try to use the tower property by conditioning on $N$ to formalize all this.

$$G_Z(s) = E[s^Z] = E[E[s^Z \mid N]] = E[E[s^{X_1} \cdots s^{X_N} \mid N]] = E[(1-p+ps)^N]= e^{\lambda p(s-1)}.$$ It remains to identify the final PGF.