A random experiment takes r possible values, with respective probabilities p 1 , ,p r Suppose the experiment is repeated N times, where N has a Poisson distribution with parameter For k 1, ,r , let N k be the number of occurrences of outcome k In other words, if N n, then (N 1 , ,N r ) has a multinomial distribution Show that N 1 , ,N r form a sequence of independent Poisson random variables and for each k 1, ,r, Nk Pois(pk)

Question: A random experiment takes r possible values, with respective probabilities p 1 , . . . ,p r . Suppose the experiment is repeated N

A random experiment takes r possible values, with respective probabilities p₁, . . . ,p_r. Suppose the experiment is repeated N times, where N has a Poisson distribution with parameter λ. For k = 1, . . . ,r , let N_k be the number of occurrences of outcome k. In other words, if N = n, then (N₁, . . . ,N_r) has a multinomial distribution.

Show that N₁, . . . ,N_r form a sequence of independent Poisson random variables and for each k = 1, . . . ,r, Nk ∼ Pois(λpk).

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