Generating function

Question

We define $Z_i=\max\{X_i,X_i'\}$ where $X_i$ and $X_i'$ are i.i.d. random variables. We would like to know the generating function of $Z_i$ in terms of the generating function of $X_i$, which is known.

Answer 1

If $p_n=\mathrm P(X=n)=\mathrm P(X'=n)$ for every $n\geqslant0$, then, for every $|s|\leqslant1$, $$ \mathrm E(s^{\max(X,X')})=\sum\limits_{n=0}^{+\infty}p_n\cdot\left(p_n+2\sum\limits_{k=0}^{n-1}p_k\right)\cdot s^n. $$