QUESTION 5 (a) Let Y and Y have a bivariate normal distribution. Show that the conditional...
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QUESTION 5 (a) Let Y₁ and Y₂ have a bivariate normal distribution. Show that the conditional distribution of Y₁ given that Y₂ = 1/₂ is a normal distribution with mean +(2-₂) and variance (1-²). (10) X (b) Consider the following bivariate joint probability density function for the random vector [[M] fxy(x, y) 1 2√1-p²a,ay +(-)"}]. exp ²1-₁0-²7 2 (1-²) {(1 2( S(x-μ1₂) ² (₁-₂)(x₂-H₂) 2p- where a > 0, 0, >0, and p < 1. Then fxy(x, y) is such that =[₁ μ= (Μx My) Σεx2 == - pa,ay pagay 0² "]. Given that Jr.x. (01, 12) = exp[-(20² + 12 + 2/192 − 22y1 − 1492 + +65}], [24] is the density of a bivariate normal random vector Y = (Y₁ Y₂ ), find (1) E[Y]. (ii) Var[Y] and (iii) k. 1847/48 -8<1/2<8 (3) (5) (6) QUESTION 5 (a) Let Y₁ and Y₂ have a bivariate normal distribution. Show that the conditional distribution of Y₁ given that Y₂ = 1/₂ is a normal distribution with mean +(2-₂) and variance (1-²). (10) X (b) Consider the following bivariate joint probability density function for the random vector [[M] fxy(x, y) 1 2√1-p²a,ay +(-)"}]. exp ²1-₁0-²7 2 (1-²) {(1 2( S(x-μ1₂) ² (₁-₂)(x₂-H₂) 2p- where a > 0, 0, >0, and p < 1. Then fxy(x, y) is such that =[₁ μ= (Μx My) Σεx2 == - pa,ay pagay 0² "]. Given that Jr.x. (01, 12) = exp[-(20² + 12 + 2/192 − 22y1 − 1492 + +65}], [24] is the density of a bivariate normal random vector Y = (Y₁ Y₂ ), find (1) E[Y]. (ii) Var[Y] and (iii) k. 1847/48 -8<1/2<8 (3) (5) (6)
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Related Book For
Probability and Statistics
ISBN: 978-0321500465
4th edition
Authors: Morris H. DeGroot, Mark J. Schervish
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