Let X be a random variable having a distribution with parameter p, 0 < p < 1, with probability function given by
p(x) = P(X = x) = (1 - p)^{x-1}p, x = 1, 2, 3, ...
We define the following function
m(t) = pe^t / (1 - qe^t), where t < -ln(1 - p).
a. Show that m(t) is the moment generating function (mgf) of X.
b. Using the mgf, find E(X), the expectation of X.
c. Let the mgf of Y be
g_Y(t) = 0.2e^t / (1 - 0.8e^t), where t < -ln(0.8).
Find the probability function p(y) of Y.