Answer the following questions: a. Let X and Y follow the bivariate normal distribution with parameters...

 

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Answer the following questions: a. Let X and Y follow the bivariate normal distribution with parameters μ₁, μ2, 01, 02, p. Show that W = X-μ₁ and Q = (Y−µ₂) - p2(X-μ₁) are independent normal random variables. b. Let X₁ and X2 be two independent normal random variables with mean zero and variance 1. Show that the vector Z (Z₁, Z2)', where Z₁ Z₂ = μ +σX, με + p62X1 + 62/1 - pX2 follows the bivariate normal distribution with parameters 1, 2, 01, 02, p. c. Let X1, X2, X3 be i.i.d. random variables N(0, 1). Show that Y₁ X₁ + 6X3 and Y₂ X2 + 8X3 have bivariate normal distribution. Find the value of 8 so that the correlation coefficient between Y₁ and Y₂ is p = 1. = = Answer the following questions: a. Let X and Y follow the bivariate normal distribution with parameters μ₁, μ2, 01, 02, p. Show that W = X-μ₁ and Q = (Y−µ₂) - p2(X-μ₁) are independent normal random variables. b. Let X₁ and X2 be two independent normal random variables with mean zero and variance 1. Show that the vector Z (Z₁, Z2)', where Z₁ Z₂ = μ +σX, με + p62X1 + 62/1 - pX2 follows the bivariate normal distribution with parameters 1, 2, 01, 02, p. c. Let X1, X2, X3 be i.i.d. random variables N(0, 1). Show that Y₁ X₁ + 6X3 and Y₂ X2 + 8X3 have bivariate normal distribution. Find the value of 8 so that the correlation coefficient between Y₁ and Y₂ is p = 1. = =

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a The joint pdf of XY is 1 2 V1 fxy x y where x f Transform XY WQ Px y 1 1 ... View the full answer