1. Let X, Y, Z follow a trinomial distribution with success probabilities px, py, Pz > 0 such that px + py + pz = 1 and sample size n E N. (a) Write down the expected values ux = E[X'] and My = E[Y]. [2] (b) Find the covariances Cov(X, X), Cov(X, Y) and Cov(Y, Y). [3] P: STAT0005, 2021-2022 14 (c) Let V, W follow a bivariate normal distribution such that E[V] = E[X], E[W] = E[Y], Var(V) = Var(X), Var(W) = Var(Y) and Cov(V, W) = Cov(X, Y). For k E {0, 1, 2, ..., n}, compute the expected value E[X|Y = ] as well as the expected value E[V|W = k] and compare them. [4] (d) For k E {0, 1, 2, ..., n}, compute the conditional variances a = Var(X Y = k) and b = Var(V|W = k). Consider whether or not a and b can be equal, as follows: i. If you think that equality of a and b is independent of the values k, py, n, write down the conditions under which the normal distribution can be used as an approximation to the binomial distribution and discuss whether they are satisfied for the conditional distribution of V given W = k. ii. If you think that equality of a and b depends on the values of k, py, n, identify the values of k, py, n for which the variances agree and comment on them. [6]