Two friends A and B go on a date and agree to meet at 7pm. Friend A is an impatient person and will scold b if A arrives before B and B arrives later than 7.15pm. Let X denote the number of minutes between 6.30 and 7.30pm that A arrives and Y denote the number of minutes 6.30 and 7.30pm that B arrives. Assume that X and Y are independent and that they are both uniformly distributed on the interval (?30, 30). (a) Draw a graph with X and Y as the two axes and shade the region which represents the event that B will be scolded. (b) Hence, or otherwise, find the probability that B will be scolded. 2. Let X and Y be two independent random variables with means µ1 and µ2 and variances ? 2 1 and ? 2 2 , respectively. (a) Show that X ? Y and X + Y are uncorrelated if and only if ? 2 1 = ? 2 2 . (b) Show that Cov(X, XY ) = µ2? 2 1 . (c) Show that Var(XY ) = ? 2 1? 2 2 + µ 2 1? 2 2 + µ 2 2? 2 1 . 3. Suppose the joint probability density function of X and Y is given by f(x, y) = x + y 50 if x = 1, 2, . . . , y and y = 1, 2, 3, 4. (a) Find the following probabilities: i. P(X ? 2, Y ? 3). ii. P(X + Y ? 4). (b) Find the marginal probability density function X and Y . (c) Find the conditional probability density function of Y given X = x and use it to find E(Y |X = 2). 4. If X and Y have a joint probability density function f(x, y) = ( 1, if 0
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