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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.

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What do you mean by i.i.d.? –  Rijul Saini Jul 31 '12 at 13:38
    
i.i.d. means 'independent identical distribution' –  Rocío Jul 31 '12 at 14:01
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If $p_n=\mathrm P(X=n)=\mathrm P(X'=n)$ for every $n\geqslant0$, then, for every $|s|\leqslant1$, $$ \mathrm E(s^{\max(X,X')})=\sum\limits_{n=0}^{+\infty}p_n\cdot\left(p_n+2\sum\limits_{k=0}^{n-1}p_k\right)\cdot s^n. $$

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No, this is the generating function of $Z$ (often denoted by $G_Z$), expressed in terms of the common probability mass function $(p_n)_n$ of $X$ and $X'$. –  Did Jul 31 '12 at 14:09
    
This is the probability mass function of the variable Z, doesn't it? And then, you have included it into the definition of the distribution function. But I am asking about the generating function of Z; that is, which relation exists between $G(P_X)$ and $G(P_X′)$, where $G(P_X)$ is the generating function of the probability distribution of the variable X. Anyway, thank you for your answer –  Rocío Jul 31 '12 at 14:10
    
I'm sorry, I haven't expressed well. I would like a relation between both generating functions, but not in terms of the probability mass function. –  Rocío Jul 31 '12 at 14:13
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@Rocío: the GF and PMF (generating function and probability mass function) are univocally related (given one of can get the other), so this relation gives you "in principle" what you want (just replace $p_n = G^{(n)}(0)/n!$). Of course, you'd prefer a simpler form, but I doubt you'll get that. –  leonbloy Jul 31 '12 at 14:58
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