Let the random variable X follow a geometric distribution, where X represents the number of trials until the first success in a sequence of independent Bernoulli trials with P(success) = p on any trial. The probability function for X is fx(x) = (1-p)*-1p, for x = 1, 2, 3, .... ; 0, otherwise. Note: In this form, you can "visualize" what's happening here. We will try something until we succeed for the first time. We are assuming that the trials are independent, and that the probability of success remains constant from trial to trial. So, there must be x-1 failures (each occurring with probability 1 - p) before our first success (occurs with probability, p), and then we stop. a. Determine the moment generating function, M.(t) for X, and use the mgf to find Kx(t), the cumulant generating function for X. [5] b. Now, using either Mx(t) or Kx(t), show that E(X) = = and Var(X) = (1-P) P p2