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2838Sum of Geometric CountsLet X 1,\dots,X r be i.i.d. geometric random variables on \ 0,1,2,\dots\ with \[ P(X i=k)=p(1-p) k. \] Use PGFs to identify the distribution of S=X 1+\cdots+X r and compute E[S].概率中等derivation未尝试面试订阅2840Poisson Number of Geometric Ticket GeneratorsLet N\sim Poisson (3). Conditional on N, let \[ S=\sum i=1 N B i, \] where each B i is geometric on \ 0,1,2,\dots\ with success parameter p=1/4, so P(B i=k)=\frac14(\frac34) k. Find the PGF of S and compute E[S].概率中等derivation未尝试面试订阅