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Consider n independent trials each resulting in any one of

Consider n independent trials, each resulting in any one of r possible outcomes with probabilities P1, P2, . . . , Pr. Let X denote the number of outcomes that never occur in any of the trials. Find E[X] and show that, among all probability vectors P1, . . . ,Pr, E[X] is minimized when Pi = 1/r, i = 1, . . . , r.

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