# Suppose a trial can result in precisely one of k mutually exclusive eventsA 1 , ,A k with probabilities p 1 , , p k , respectively, where P 1 + + P k = 1. Suppose that n independent trials are performed. Show that the probability of getting x 1 A 1

Chapter 24, PROBLEM SET 24.7 #17

Suppose a trial can result in precisely one of k mutually exclusive events A_{1}, · · ·,A_{k} with probabilities p_{1}, · · ·, p_{k}, respectively, where P_{1} + · · · + P_{k} = 1. Suppose that n independent trials are performed. Show that the probability of getting x_{1}A_{1}’s, · · ·, x_{k}A_{k}'s is

where 0 ≤ x_{j }≤ n, j = 1, · · ·,k, and x_{1} + · · · + x_{k} = n. The distribution having this probability function is called the multinomial distribution.

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