9. (10 points) (i) For bivariate discrete r.v.'s (X, Y), show that Var (W) = azo'x +boy + 2aboxy based on E [W] = aux + buy, where W = aX + by. (ii) For bivariate discrete r.v.'s (X, Y), show that if X and Y are independent, then Cov (X, Y) = 0. (iii) Exercise 4.77 in SBE on page 193.Application Exercises 4.77 A researcher suspected that the number of between- meal snacks eaten by students in a day during final examinations might depend on the number of tests a student had to take on that day. The accompany- ing table shows joint probabilities, estimated from a survey. Number of Number of Tests (X) Snacks (Y) 0 1 2 3 0 0.07 0.09 0.06 0.01 0.07 0.06 0.07 0.01 W N H 0.06 0.07 0.14 0.03 0.02 0.04 0.16 0.04 a. Find the probability distribution of X and compute the mean number of tests taken by students on that day. b. Find the probability distribution of Y and, hence, the mean number of snacks eaten by students on that day. c. Find and interpret the conditional probability dis- tribution of Y, given that X = 3. d. Find the covariance between X and Y. e. Are number of snacks and number of tests indepen- dent of each other