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Assignment 10

Assignment 10 - Stat 5101(Geyer Fall 2009 Homework...

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Stat 5101 (Geyer) Fall 2009 Homework Assignment 10 Due Wednesday, December 2, 2009 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 10-1. Suppose ( X 1 , X 2 ) is a bivariate normal random vector, and assume it is nondegenerate. Write its PDF in terms of the mean vector and variance matrix. Then rewrite its PDF in terms of new parameters, which are E ( X 1 ) = μ 1 E ( X 2 ) = μ 2 sd( X 1 ) = σ 1 sd( X 2 ) = σ 2 cov( X 1 , X 2 ) = ρσ 1 σ 2 Then simplify your expression for the PDF so it contains no matrices, no matrix inverses, determinants, or matrix multiplication. Hint: the inverse of a matrix A = a 11 a 12 a 21 a 22 can be done by Cramer’s rule obtaining A - 1 = 1 det( A ) a 22 - a 12 - a 21 a 11 assuming A is invertible, which it is if det( A ) is not zero. 10-2. Suppose ( X 1 , X 2 ) is a nondegenerate bivariate normal random vec- tor. Calculate the conditional PDF of X 1 given X 2 not using the theory developed in class. Just use conditional = joint

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Assignment 10 - Stat 5101(Geyer Fall 2009 Homework...

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