5. Consider n independent rolls of a k-sided fair die with k 2 2: the sides of the die are labelled 1, 2, ..., k and each side has probability 1/k of facing up after a roll. Let the random variable X; denote the number of rolls that result in side i facing up. The joint p.m.f. of X1, ..., Xx is multinomial. (The multinomial distribution was discussed in problem 3 of recitation 4.) (a) Which of the following statements is always correct? Try to answer without doing any calculations. (i) X1 and X2 are uncorrelated. (ii) X1 and X2 are positively correlated. (iii) X1 and X2 are negatively correlated. (b) Find the covariance, cov(X1, X2), of X, and X2. Express your answer as a function of n and k. Hint: Use indicator variables to encode the result of each roll. (c) Suppose now that the die is biased, with a probability p; 0 that the result of any given die roll is i, for i = 1, 2, ..., k. We still consider n independent tosses of this biased die and define X, to be the number of rolls that result in side i facing up. Generalize your answer to part b: Find cov( X1, X2) for this case of a biased die. Express your answer as a function of n, k, p1, p2