16.3 Exercise 6 — Applied Probability

Consider the composite demand D introduced in the section on Random Sums in "Random Selecton"

D = \sum_{n = 0}^{\infty} I_{\left\{k\right\}} \left(N\right) X_{n} \text{where} X_{n} = \sum_{k = 0}^{n} Y_{k} , Y_{0} = 0

Suppose $\left\{N , H\right\}$ is independent, $\left\{N , Y_{i}\right\} ci \left|\right. H$ for all i, and $E \left[\right. Y_{i} \left|\right. H \left]\right. = e \left(H\right)$ , invariant with i. Show that $E \left[\right. D \left|\right. H \left]\right. = E \left[N\right] E \left[\right. Y \left|\right. H \left]\right. a . s .$ .

Pfeiffer, Applied Probability, 16.3 Problems on Conditional Independence, Given a Random Vector, exercise 6

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