STAT 400 Discussion 8 Spring 2015 1. Suppose that the random variables X and Y have joint p.d.f. f ( x, y ) given by f ( x, y ) = C x 2 y, 0 < x < y, x + y < 2, zero elsewhere. a) Sketch the support of ( X , Y ). That is, sketch { 0 < x < y, x + y < 2 }. b) What must the value of C be so that f ( x, y ) is a valid joint p.d.f.? c) Find P ( Y < 2 X ). d) Find P ( X + Y < 1 ). e) Find the marginal probability density function for X. f) Find the marginal probability density function for Y. “Hint”: Consider two cases: 0 < y < 1 and 1 < y < 2. g) Find E ( X ). h) Find E ( Y ). i) Find E ( X Y ).2. Let X and Y have the joint probability density function f ( x, y ) = C, 0 x 1, 0 y x ( 1 – x ), zero elsewhere. ( That is, suppose that ( X, Y ) is uniformly distributed over the region defined by 0 x 1, 0 y x ( 1 – x ). ) a) Find the value of C so that f ( x, y ) is a valid joint p.d.f. b) Find the marginal probability density function of X, f X ( x ). c) Find the marginal probability density function of Y, f Y ( y ). d) Are X and Y independent? If not, find Cov ( X, Y ). 3. Every month, the government of Neverland spends X million dollars purchasing guns and Y million dollars purchasing butter. Assume X and Y are independent, X has a Normal distribution with mean 265 and standard deviation 40 (in millions of dollars), and Y has a Normal distribution with mean 170 and standard deviation 30 (in millions of dollars). a) Find the probability that the government of Neverland spends more on guns than on butter during a given month. That is, find P ( X > Y ). b) Find the probability that the government of Neverland spends more on guns than twice the amount it spends on butter during a given month. That is, find P ( X > 2 Y ). c) Find the probability that the government of Neverland exceeds the 500-million spending limit during a given month. That is, find P ( X + Y > 500 ).1. Suppose that the random variables X and Y have joint p.d.f. f ( x, y ) given by f ( x, y ) = C x 2 y, 0 < x < y, x + y < 2, zero elsewhere. a) Sketch the support of ( X , Y ). That is, sketch { 0 < x < y, x + y < 2 }. b) What must the value of C be so that f ( x, y ) is a valid joint p.d.f.? Must have dydxyxf , = 1. 10 2 2 dxdyyxxxC = 10222 2 dxyxxyxyC = 10 22 2 2 2 dxxx xC = 1032 2 2 dxxx CC = 0143 2 3 2 xxCC = 6C = 1. C = 6.c) Find P ( Y < 2 X ). x + y = 2 & y = 2 x x = 32, y = 34. 1 – 320 22 2 6 dxdyyxxx = 1 – 3202222 3 dxyxxyxy = 1 – 320 2 2 2 4 2 3 dxxx x = 1 – 320432 9 12 12 dxxxx = 1 – 032543 59 3 4 xxx = 543 32 5932 332 41 = 13587. OR 320 2 2 6 dxdyyxxx + 132 2 2 6 dxdyyxxx = … OR 10 2 2 6 dydxyxyy + 341 22 2 6 dydxyxyy = … OR 1 – 340 20 2 6 dydxyxy – 234 20 2 6 dydxyxy = …d) Find P ( X + Y < 1 ). 5.00 1 2 6 dxdyyxxx = 5.00122 3 dxyxxyxy = 5.00 22 2 1 3 dxxx x = 5.0032 6 3 dxxx = 05.043 23 xx = 43 21 2321 = 32381 = 321 = 0.03125. e) Find the marginal probability density function for X. First, X can only take values in ( 0 , 1 ). f X ( x ) = , dyyxf = xxdyyx2 2 6 = 3222 xyxyyx = 222 2 3 xx x = 12 x 2 – 12 x 3 = 12 x 2 ( 1 – x ), 0 < x < 1.f) Find the marginal probability density function for Y. “Hint”: Consider two cases: 0 < y < 1 and 1 < y < 2. First, Y can only take values in ( 0 , 2 ). f Y ( y ) = , dxyxf = 216106 20 2 0 2 ydxyxydxyxyy = 21 210 2 02303 yyxyyxxyxxyx = 212 210 2 34 yyyyy g) Find E ( X ). E ( X ) = X dxxfx = 10 2 112 dxxxx = 0.60. h) Find E ( Y ). E ( Y ) = Y dyyfy = 104 2 dyyy + 213 22 dxyyy = 151131 = 1516. i) Find E ( X Y ). E ( X Y ) = 10 2 2 6 dxdyyxyxxx = … = 3522.2. Let X and Y have the joint probability density function f ( x, y ) = C, 0 x 1, 0 y x ( 1 – x ), zero elsewhere. ( That is, suppose that ( X, Y ) is uniformly distributed over the region defined by 0 x 1, 0 y x ( 1 – x ). ) a) Find the value of C so that f ( x, y ) is a valid joint p.d.f. Must have 1 = 10 10 dxdyxxC = 10 2 dxxxC = 6C. C = 6. b) Find the marginal probability density function of X, f X ( x ). f X ( x ) = 10 6xxdy = 1 6 xx , 0 < x < 1. c) Find the marginal probability density function of Y, f Y ( y ). y x ( 1 – x ) x 1 < x < x 2 , where x 1 = y4121 , x 2 = y4121 . f Y ( y ) = 2 1 6 xxdx = 12 6 xxx = y41 12 = y41 6 , 0 < y < 41.d) Are X and Y independent? If not, find Cov ( X, Y ). f ( x, y …
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