1. The problem statement, all variables and given/known data Let X be a random variable with support on the positive integers (1, 2, 3, . . .) and PMF f(x) = C2 ^(-x) . (a) For what value(s) of C is f a valid PMF? (b) Show that the moment generating function of X is m(t) = Ce^t/(2− e^t) , and determine the interval for t for which it is valid. (You may use your value for C calculated in question 1, if you would like). (c) Using the MGF, calculate the expected value and the variance of X. 3. The attempt at a solution a) sum from -∞ to ∞ of C/(2^x)=1 C(1/2+1/4.....)=1 C=1 as it converges b) m(t)=E[e^tx]=integral from -∞ to ∞ of ((e^tx)*(2^(-x))) =integral from -∞ to ∞ of (e^tx)/(2^x) is this the right way to go about calculating it?