t15 Man-ts} Suppose U N Bernoulli{p1} and 1" ~ Bernoullilpg} are independent random variables. Consider the transformation V = Y and the resulting sum of W = U + V. a) Derive the probability function of V = Y. b} Derive the probability function of W = U + V using the following two methods: (i) Derive the probability function, fw , directly by carefully considering the out- comes of U and V. (ii) Use the method of probability generating functions to derive fw. That is, rst compute Gu[z) and Gp(z), the probability generating functions of U and V respectively. Then using GU and Gv, compute the probability generating function of W and then extract the probability function of W. 1:} Show that y is a valid probability function and plot the cumulative distribution function Fw(w). :1) Using the pgf in b} ii) derive the moment generating function (mgf) of W. Then using the mgf compute the mean and variance of W. Note: You don"t need to simplify