Question

Find the (probability generating function) p.g.f.’s of the following distributions:[3+3=6]

•P(X=x) =(exp(−λ)λ^x)/((1−exp(−λ))x!) , for x= 1,2,3,..., and λ >0.

•P(X=x) =((pq)^x)(1−q^(N+1))^−1,for x= 0,1,..., N; where 0 < p < 1, p+q= 1.

Answer 1

If f(x) is any discrete distribution of X, then the Probability Generating function of X is given by -

P(t) = E(t^x) , where t is a continuous real valued variable.

Answer to 1st Question:

Therefore,

The Probability Generating function is P(t) = {exp(t) - 1} / {exp() - 1}

Answer to 2nd Question:

Therefore,

The Probability Generating Function is P(t) = {1 - (pqt)^(N + 1)} / {(1 - q^(N + 1)) (1 - pqt)}