I have two independent variables $X\sim \mathcal B(n,p)$, Binomial and $Y\sim \mathcal P(\lambda)$, Poisson. How would I go about finding the distribution of $Z=XY$ and the couple $(Z,S)$, where $S=X+Y$?
Product of two independent stochastic variables $XY$
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probability
probability-theory
probability-distributions
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0See Edit. $ $ $ $ – 2012-11-09
1 Answers
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Note that $(Z,S)$ is almost surely in a subset $D_0\cup D_1$ of $\mathbb Z\times\mathbb Z$, defined by $ D_0=\{(n_0(s,t),2s)\mid 0\leqslant t\leqslant s\},\qquad n_0(s,t)=s^2-t^2, $ and $ D_1=\{(n_1(s,t),2s+1)\mid 0\leqslant t\leqslant s\},\qquad n_1(s,t)=s^2+s-t^2-t. $ Then, for every $s\geqslant0$ and $0\leqslant t\leqslant s$, $ [Z=n_0(s,t),S=2s]=[X=s-t,Y=s+t]\cup[X=s+t,Y=s-t], $ and $ [Z=n_1(s,t),S=2s+1]=[X=s-t,Y=s+t+1]\cup[X=s+t+1,Y=s-t], $ from which explicit formulas can be deduced.