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HW8 - Mathematics Department UCLA T Richthammer spring 09...

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Mathematics Department, UCLA T. Richthammer spring 09, sheet 8 May 15, 2009 Homework assignments: Math 170B Probability, Sec. 1 78. Let Z 1 , Z 2 be independent standard normal, X 1 = Z 1 + Z 2 +1, X 2 = Z 1 - Z 2 - 1. Calculate the expectation vector and the covariance matrix of X 1 , X 2 . Are X 1 , X 2 independent? Answer: We have A = ( 1 1 1 - 1 ) and b = ( 1 - 1 ) . By the lecture b is the expectation vector and the covariance matrix is C = AA t = ( 1 1 1 - 1 )( 1 1 1 - 1 ) = ( 2 0 0 2 ) . This shows that X 1 , X 2 are uncorrelated and thus independent. 79. Let X, Y have a bivariate normal distribution with common mean 5, common standard de- viation 2, and correlation 0.5. Calculate P ( X + Y 20), and explain without a calculation whether this probability increases or decreases when the correlation increases. Answer: Z = X + Y is a normal RV with E ( Z ) = E ( X ) + E ( Y ) = 10 and V ( Z ) = V ( X ) + V ( Y ) + 2 Cov ( X, Y ) = 4 + 4 + 4 = 12, so P ( Z 20) = P ( Z - 10 12 10 12 ) 1 - Φ(2 . 89) 0 . 0019. This probability increases when the correlation increases (since then X, Y like to be large at the same time). 80. In both cases show that the RVs X have the same joint distribution as the RVs X : (a) Let Z 1 , Z 2 , Z 3 be independent standard normal RVs and let X = ( X 1 X 2 ) = A ( Z 1 Z 2 Z 3 ) and X = ( X 1 X 2 ) = A ( Z 1 Z 2 ) , where A = ( 1 2 1 2 0 1 2 0 1 2 ) and A = ( 1 0 1 2 3 2 ) . (b) Let Z 1 , . . . , Z n be independent standard normal RVs, A R m × n and b R m , and let B R n × n be orthogonal. Consider X = AZ + b and X = ABZ + b . Answer: If X and X are multivariate normal, they have the same distribution iff they have the same expectation vector and covariance matrix. In both (a) and (b) they have the same expectation vector, so it suffices to check that the covariance matrices are the same. (a) C = AA t = ( 1 1 2 1 2 1 ) and C = A ( A ) t = ( 1 1 2 1 2 1 ) . (b) C = AA t and C = ( AB )( AB ) t = ABB t A t = AA t = C . 81. Let X, Z be independent RVs such that X is standard normal and Z has the PMF p ( - 1) = p (1) = 1 2 . Let Y = XZ .
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