138. Let X be a discrete rv with possible values 0, 1, 2, . . . or some subset of these. The function is called the probability generating function [e.g., h(2)  2xp(x), h(3.7)  (3.7)xp(x), etc.].

a. Suppose X is the number of children born to a family, and p(0)  .2, p(1)  .5, and p(2)  .3.

Determine the pgf of X.

b. Determine the pgf when X has a Poisson distribution with parameter l.

c. Show that h(1)  1.

d. Show that (assuming that the derivative can be brought inside the summation, which is justi ed). What results from taking the second derivative with respect to s and evaluating at s  0? The third derivative? Explain how successive differentiation of h(s) and evaluation at s  0 generates the probabilities in the distribution. Use this to recapture the probabilities of

(a) from the pgf. Note: This h¿1s 2 0s0  p112 shows that the pgf contains all the information about the distribution knowing h(s) is equivalent to knowing p(x).