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In a population of four classes of individuals, the proportion of individuals of Class i is pi for i = 1, 2, 3, 4. Suppose you sample independently from this population, and suppose the sample size is a Poisson random variable with parameter n for a fixed positive integer n.
For 1 ≤ i ≤ 4, let Ni be the number of Class i people in the sample.
a) What is the joint distribution of N1, N2, N3, N4?
b) Find the probability that the sample contains at least one individual from each class.
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