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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.?2012-07-31
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    i.i.d. means 'independent identical distribution'2012-07-31

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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 $X'$.2012-07-31
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    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 answer2012-07-31
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    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.2012-07-31
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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.2012-07-31