An elementary experiment is independently performed N times where N is a Poisson rv of mean . Let {a 1 , a 2 , . . . , a K } be the set of sample points of the elementary experiment and let pk, 1 ≤ k...

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

a) Let N_k
denote the number of elementary experiments performed for which the output is a_k. Find the PMF for N_k
(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.