Suppose a vendor stocks four types of pens, each sold for $2, $4, $8, and $10. Suppose X keeps track of revenue from a single sale, and we know that X has the following distribution: T 2 4 8 10 P(X = x) .2 .2 .1 .5Y, the sample mean for a random sample of size two coming from the distribution for X. Z, the sample mean for a random sample of size three coming from the distribution for X. (a) Calculate the moment generating function for X, that is M X (t) = Elk\" ). (b) Cacnlate the moment generating function for Y, that is My (t) = E[cw), using the distribution table for Y. lCompare this with what we expect from the theorem stated in class, that is My [t] = (MX (%})2. (c) Cacnlate the moment generating mction for Z, that is Mz] = 113(th ), using the distribution table for Z. Compare this with what we expect from the theorem stated in class, that is Mg) = (MX (9)3. (d) Let K = 5X + 2, calculate the probability distribution table for the random variable K. Now calculate the mgf for K, that is My (t) using this distribution table. Verify that Mt) = emMth)