An elementary experiment is independently performed N times where N is a Poisson rv of mean . Let…

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An elementary experiment is independently performed N times where N is a Poisson rv of mean . Let {a1, a2, . . . , aK} be the set of sample points of the elementary experiment and let pk, 1 ≤ k ≤ K, denote the probability of ak.

a) Let Nk denote the number of elementary experiments performed for which the output is ak. Find the PMF for Nk (1 ≤ k ≤ K).

b) Find the PMF for N1 + N2.

c) Find the conditional PMF for N1 given that N = n.

d) Find the conditional PMF for N1 + N2 given that N = n.

e) Find the conditional PMF for N given that N1 = n1.

 

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