$X_1,X_2,...\sim Pois(7), $ and independent random variables. $Y \sim Geom(1/4)$ independent from the $X_i$.

My question is the characteristic function of: $X_1+X_2+...+X_Y$

Can someone tell me how to do this.

I could do it if the sum went to some $n$, but I have no idea how to this.


Let $Z=g(Y,X_1,X_2 \cdots)=\sum_{i=1}^Y X_i$. The characteristic function is $$\Phi_Z(t)= E[\exp (i t Z)]=E[E[\exp (i t Z)|Y]]$$ where the later is a consecuence of the tower property: $E[g(X,Y)]=E[E[g(X,Y)|Y]]$


$$\Phi_Z(t)=E[ E[\prod_{i=1}^Y \exp{( i t X_i)}|Y]]=E[ \Phi_X(t)^Y]$$

where $\Phi_X(t)$ is the CF of a Poisson (we've used here the independence property). What's left is to evaluate the expression in brackets and compute its expectation with respect to $Y$.


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