STAT 400 Lecture AL1 Practice Problems 4 Spring 2015 Dalpiaz 1. Let X and Y have the joint p.d.f. f X , Y ( x, y ) = C x 2 y 3, 0 < x < 1, 0 < y < x , zero elsewhere. a) What must the value of C be so that f X , Y ( x, y ) is a valid joint p.d.f.? b) Find P ( X + Y < 1 ). c) Let 0 < a < 1. Find P ( Y < a X ). d) Let a > 1. Find P ( Y < a X ). e) Let 0 < a < 1. Find P ( X Y < a ). f) Find f X ( x ). g) Find E ( X ). h) Find f Y ( y ). i) Find E ( Y ). j) Find E ( X Y ). k) Find Cov ( X, Y ). l) Are X and Y independent? 2. Let X and Y have the joint probability density function f X , Y ( x, y ) = x1, x > 1, 0 < y < x1, zero elsewhere. a) Find f X ( x ). b) Find E ( X ). c) Find f Y ( y ). d) Find E ( Y ). 3. Let X and Y have the joint probability density function f X , Y ( x, y ) = x1, 0 < x < 1, 0 < y < x, zero elsewhere. a) Find f X ( x ). b) Find E ( X ). c) Find f Y ( y ). d) Find E ( Y ). e) Find P ( X + Y ≥ 1 ). f) Find Cov ( X, Y ).4. Let X and Y be two random variables with joint p.d.f. f ( x, y ) = 64 x exp { – 4 y } = 64 yex4 −, 0 < x < y < ∞, zero elsewhere. a) Find P ( X 2 > Y ). b) Find the marginal p.d.f. f X ( x ) of X. c) Find the marginal p.d.f. f Y ( y ) of Y. d) Are X and Y independent? If not, find Cov ( X, Y ) and ρ = Corr ( X, Y ). e) Let a > 1. Find P ( Y > a X ). f) Let a > 0. Find P ( X + Y < a ). 5. Let the joint probability mass function of X and Y be defined by p ( x, y ) = 32yx +, x = 1, 2, y = 1, 2, 3, 4. a) Find P ( Y > X ). b) Find p X ( x ), the marginal p.m.f. of X. c) Find p Y ( y ), the marginal p.m.f. of Y. d) Are X and Y independent? If not, find Cov ( X, Y ). 6. Let the joint probability mass function of X and Y be defined by p ( x, y ) = 30yx⋅, x = 1, 2, y = 1, 2, 3, 4. a) Find P ( Y > X ). b) Find p X ( x ), the marginal p.m.f. of X. c) Find p Y ( y ), the marginal p.m.f. of Y. d) Are X and Y independent? If not, find Cov ( X, Y ).7. Suppose the joint probability density function of ( X , Y ) is ()≤≤≤=otherwise010 ,2xyyxCyxf a) Find the value of C that would make ( )yxf , a valid probability density function. 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) Find P ( X > 2 Y ). e) Find P ( X + Y < 1 ). f) Are X and Y independent? If not, find Cov ( X, Y ). 8. Let X and Y have the joint probability density function f ( x, y ) = C x, 0 ≤ x ≤ 1, 0 ≤ y ≤ x ( 1 – x ), zero elsewhere. a) Find the value of C so that f ( x, y ) is a valid joint p.d.f. b) Find f X ( x ). c) Find E ( X ). d) Find f Y ( y ). e) Find E ( Y ). f) Are X and Y independent? 9. Suppose that ( X, Y ) is uniformly distributed over the region defined by x ≥ 0, y ≥ 0, x 2 + y 2 ≤ 1. That is, f ( x, y ) = C, x ≥ 0, y ≥ 0, x 2 + y 2 ≤ 1, zero elsewhere. a) What is the joint probability density function of X and Y ? That is, find the value of C so that f ( x, y ) is a valid joint p.d.f. b) Find P ( Y > 2 X ). c) Find P ( X + Y < 1 ). d) Are X and Y independent?1. Let X and Y have the joint p.d.f. f X , Y ( x, y ) = C x 2 y 3, 0 < x < 1, 0 < y < x , zero elsewhere. a) What must the value of C be so that f X , Y ( x, y ) is a valid joint p.d.f.? ∫ ∫10 03 2 dxdyyxxC = ∫104 4 dxxC = 20C = 1. ⇒ C = 20. b) Find P ( X + Y < 1 ). y = x and y = 1 – x x = y 2 and x = 1 – y ⇒ y = 215 −. P ( X + Y < 1 ) = ∫ ∫−−215013 2 20 2 dydxyxyy = ( )∫−−−21509 3 3 3201 320 dyyyy = ∫−−−+−21509 6 5 4 3 3203202020 320 dyyyyyy= 021510 7 6 5 4 3221203104 35 −−−+− yyyyy ≈ 0.030022. OR y < x and y = 1 – x ⇒ x = 2532151215 2−=−−=−. P ( X + Y < 1 ) = ∫ ∫−−−125313 2 20 1 dxdyyxxx = ( )( )∫−−−−125342 4 1 55 1 dxxxx = ( )∫−−+−+−−12536 5 4 3 2 5202520 5 1 dyxxxxx = 25317 6 5 4 3 7531055 35 1−−+−+−− xxxxx ≈ 0.030022. c) Let 0 < a < 1. Find P ( Y < a X ). P ( Y < a X ) = ∫ ∫10 03 2 20 dxdyyxxa = ∫106 4 5 dxxa = 4 75a.d) Let a > 1. Find P ( Y < a X ). y = x and y = a x ⇒ x = 2 1a, y = a1. P ( Y < a X ) = ∫ ∫−aayydydxyx 103 2 20 12 = ∫−−adyyay 109 36 320 3 20 1 = 10 721a−. P ( Y < a X ) = ∫∫−2 103 2 20 1axxadxdyyx = ()∫−−2 1064 4 5 5 1adxxax = 10 721a−. e) Let 0 < a < 1. Find P …
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