Determine C, the generating functions GX(s), GY (s), GZ(s) and the probability mass function for the random variable Z = X + Y .

Question

Consider two independent random variables X and Y . Random variable X can take only even values 0, 2, 4, . . ., and for any integer k it takes value 2k with probability P[X = 2k] = $C/4^k$ , where C is a constant. Random variable Y is a Bernoulli random variable taking values 0 and 1 only, with P[Y = 1] = 1/3. Determine the value of C, the generating functions GX(s) and GY (s), and hence find the generating function GZ(s) and the corresponding probability mass function for the random variable Z = X + Y . Explain how the probability mass function of Z could be obtained without using generating functions