Answer to Question #334014 in Statistics and Probability for Dinor123

Question

Suppose the number of automobile accidents a driver will be involved in during a one-year period is a random variable X having a Poisson distribution with parameter θ ,where θ is a measure of accidents proneness that varies from driver to driver in accordance with Gamma distribution given by

f(θ) = (p/q)^α × θ^(α−1)/Γ(α)×exp {(−p/q) θ} , θ > 0

where α is a positive integer, p, q are positive constants and p + q = 1.

(a) show that he factorial moment generating function of x is g(s) = (p/1 − qs)^α

.

(b) Using the uniqueness property of probability generating functions, identify completely the

distribution of X.

(c) If p =2/3 and α = 12, find

E{x(x − 1)· · ·(α − k + 1)},

where k is a positive integer