1. Covariance and correlation (a) Let X ~ U ([0, a]) for some a > 0. Determine the covariance Cov[X, X²] and the correlation coefficient p(X, X²). (b) Consider rolling a fair die twice, and let X1 and X2 denote the outcome of the die on the first and second round, respectively. We define S = X1 + X2 (sum) and M = (highest outcome). max{X1, X2} (i) Write down the joint probability mass function ps,M in a table. (ii) Calculate Cov[S, M] and p(S, M).

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