stat1301hw4sol

stat1301hw4sol - 11/12 THE UNIVERSITY OF HONG KONG...

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11/12 p. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability and Statistics I Assignment 4 Solution 1. (a) () 36 1 2 , = = = i Y i X P for 6 , 5 , 4 , 3 , 2 , 1 = i 18 1 , = + = = j i Y i X P for 1 ,..., 2 , 1 = i j ; 6 , 5 , 4 , 3 , 2 , 1 = i (b) 36 , i i Y i X P = = = for 6 , 5 , 4 , 3 , 2 , 1 = i 36 1 , = = = j Y i X P for 6 ,..., 2 , 1 + + = i i j ; 6 , 5 , 4 , 3 , 2 , 1 = i (c) 36 1 , = = = i Y i X P for 6 , 5 , 4 , 3 , 2 , 1 = i 18 1 , = = = j Y i X P for 6 ,..., 2 , 1 + + = i i j ; 6 , 5 , 4 , 3 , 2 , 1 = i 2. ( ) ( ) 2 1 2 1 1 1 1 , 2 2 2 1 1 x x x x p p p p p p x X x X P + = = = = , ,... 2 , 1 , 0 1 = x , ,... 2 , 1 , 0 2 = x 3. 15 1 , 2 1 = = = j N i N P , i j = 6 ,..., 1 , 5 , 4 , 3 , 2 , 1 = i . 4. (a) 0.15, 0.2 (b) , = = = 1 35 . 0 0 65 . 0 x x x p X = = = = 2 35 . 0 1 35 . 0 0 3 . 0 y y y y p Y (c) (d) 0.35, 1.05, 0.2275, 0.6475 (e) –0.1175, –0.3061 = = = = = 3 1 . 0 2 3 . 0 1 5 . 0 0 1 . 0 w w w w w p W (f) = = = = = 2 7 2 1 7 1 0 7 4 1 | y y y X y p 5. (a) The random variables X and Y are not independent because the supports of them depends on the values of each other according to the inequality 1 < + y x . (b) () ( ) ( ) ( ) x x X y y x dy y x dy y x f x f = = = 1 0 2 1 0 2 1 6 1 6 , , 2 1 3 x = 1 0 < < x (c) () ( ) 4 1 4 3 2 2 3 3 6 3 1 3 1 0 4 3 2 1 0 3 2 1 0 2 = + = + = = x x x dx x x x dx x x X E
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11/12 p. 2 Obviously Y has the same marginal distribution as X . Therefore () 4 1 = Y E . For the variance, consider () () 10 1 5 3 2 3 3 6 3 1 3 1 0 5 4 3 1 0 4 3 2 1 0 2 2 2 = + = + = = x x x dx x x x dx x x X E 80 3 4 1 10 1 2 = = = Y Var X Var Now for the covariance, () ( ) ∫∫ = = 1 0 1 0 1 6 , y dxdy y x xy dxdy y x xyf XY E [] = 1 0 1 0 3 2 2 1 3 dy y x y y x y = 1 0 3 1 dy y y 20 1 5 4 1 1 0 5 4 1 0 3 = = = y y dy y y ( ) ( ) 80 1 4 1 4 1 20 1 = × = = Y E X E XY E XY Cov Therefore the correlation coefficient between X and Y is ( ) 3 1 80 3 80 3 80 1 , , = × = = Y Var X Var Y X Cov Y X Corr .
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stat1301hw4sol - 11/12 THE UNIVERSITY OF HONG KONG...

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