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Poisson distro

Source: LIMIT 2019 CCS2 P8

April 28, 2021
probability

Problem Statement

Let X1,X2,…X_1,X_2,\ldots be a sequence of independent random variables distributed exponentially with mean 11. Suppose N\mathbb N is a random variable independent of XiX_i's that has a Poisson distribution with mean λ>0\lambda>0. What is the expected value of X1+X2+…+XNX_1+X_2+\ldots+X_N? <spanclass=′latex−bold′>(A)</span> N2<span class='latex-bold'>(A)</span>~N^2 <spanclass=′latex−bold′>(B)</span> λ+λ2<span class='latex-bold'>(B)</span>~\lambda+\lambda^2 <spanclass=′latex−bold′>(C)</span> λ2<span class='latex-bold'>(C)</span>~\lambda^2 <spanclass=′latex−bold′>(D)</span> λ<span class='latex-bold'>(D)</span>~\lambda
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