Assignment 4 - 11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE Assignment 4 Due Date: November 21, 2011 (Hand in your solutions for Questions 5, 8, 10, 14, 17, 27, 32, 36) 1. Two fair dice are rolled. Find the joint probability mass functions of X and Y when (a) X is the largest value obtained on any die and Y is the sum of the values; (b) X is the value on the first die and Y is the larger of the two values; (c) X is the smallest and Y is the largest value obtained on the dice; 2. Consider a sequence of independent Bernoulli trials, each of which is a success with probability p . Let be the number of failures preceding the first success, and let be the number of failures between the first two successes. Find the joint mass function of and . 1 X 2 X X 1 2 X 3. A bin of 6 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by the number of tests made until the first defective is spotted and by the number of additional tests until the second defective is spotted; find the joint probability mass function of and . 1 N 1 N 2 N 2 N 4. Consider the following joint pmf of X and Y . Y X 0 1 2 0 0.1 0.3 0.25 1 0.2 0.05 0.1 Find (a) , ; () Y X P = () Y X P > (b) the marginal pmfs of X and Y ; (c) the pmf of Y X + ; (d) , , , ; () X E () Y E () X Var () Y Var (e) , ; () Y X Cov , () Y X Corr , (f) the conditional pmf of Y given 1 = X . 5. Suppose that X and Y are random variables with joint probability density function () () > > < + = otherwise 0 ; 0 , 0 , 1 for 1 6 , y x y x y x y x f (a) Are X and Y independent? Why? (b) Determine the marginal probability density function of X . (c) Determine the correlation coefficient between X and Y . (d) Determine the conditional probability density function of Y given . x X = P. 1
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11/12 6. Suppose that a point is uniformly chosen on a square of area 1 having vertices (0,0), (0,1), (1,0) and (1,1). Let X and Y be the coordinates of the point chosen. (a) Find the marginal pdfs of X and Y . (b) Are X and Y independent? (c) Find the probability that the distance from ( X , Y ) to the center of the square is greater than 4 1 . 7. The joint probability density function of X and Y is given by () ( ) < < = otherwise 0 0 , if , 2 2 y y x y e x y c y x f y . (a) Find the value of c . (b) Find the marginal densities of X and Y . (c) Find . () X E 8. The joint probability density function of X and Y is given by () < < < < + = otherwise 0 2 0 , 1 0 if 2 , 2 y x xy x c y x f . (a) Are X and Y independent? (b) Find the value of
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.

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Assignment 4 - 11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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