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.
Generating function
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combinatorics
stochastic-processes
generating-functions
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0i.i.d. means 'independent identical distribution' – 2012-07-31
1 Answers
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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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1@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