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STATISTICS 244 Problem Set 3 9:00-10:20 TTh Due Thursday January 26 1. Assume the random variable X has the Bernoulli distribution (i.e., binomial with n = 1): P(X = 1) = q = 1 – P(X = 0). (a) Find E( ). (b) Find E(X 3 ). (c) Find E(X 10 ). (d) Find Var(X 3 ). (e) Find E(3X 3 +4). (f) Find Var(3X 3 +4). 2. Consider the following bivariate distribution: Y 2 3 4 5 1 .1 .1 .0 .0 X 2 .0 .2 .2 .1 3 .0 .0 .1 .2 (a) Find the marginal distributions of X and Y. (b) Find the conditional distribution of Y given X = 1. Find the conditional distribution of Y given X = 2. (c) Find E(Y). (d) Find the covariance of X and Y. 3. Suppose an urn contains 3 tickets numbered “1”, 3 tickets numbered “2”, 2 ticket numbered “3”, and 1 ticket numbered “4”. A student draws a ticket at random and notes the number, X. The student then returns the ticket to the urn, shakes it up, and draws again, noting the number, Y. Let Z = the minimum of X and Y. (a) Find the probability distribution of X, the cumulative distribution of X, and graph both. (b) Find the probability distribution of Z. (c) Find E(X), E(Y), E(Z). (d) Find the bivariate probability function p(x, z) of X and Z, and the covariance of X and Z.

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