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Unformatted text preview: Stat 116 Homework 6 Due Wednesday, May 21st. Please show work and justify answers. No credit for a final answer with no explanation, even if the answer is correct. 1. Let X and Y be distributed std. Normal with correlation between X and Y equal to ρ . Compute the joint and marginal distributions of X + Y and X − Y , are X + Y and X − Y independent? Solution: Let u = X + Y and v = X − Y so that x = 1 2 ( u + v ) and y = 1 2 ( u − v ). Then the Jacobian is  J  = 1 2 and in the exponent of the bivariate Normal density we have − 1 2(1 − ρ 2 ) ( x 2 − 2 ρxy + y 2 ) = − 1 2(1 − ρ 2 ) ( u 2 (1 − ρ ) + v 2 (1 + ρ )) Therefore the density transforms to f U,V ( u,v ) = 1 4 π radicalbig 1 − ρ 2 exp braceleftbigg − u 2 4(1 + ρ ) − v 2 4(1 − ρ ) bracerightbigg So clearly U and V are independent since there joint density factors into marginal densities, where U ∼ N (0 , 2(1+ ρ )) and V ∼ N (0 , 2(1 − ρ )). 2. Let U ∼ χ 2 ( r ) and V ∼ χ 2 ( s ) be independent for s > 2. (a) Compute E parenleftBig U/r V/s parenrightBig . Solution: Notice U ∼ Γ( α = r/ 2 ,β = 1 / 2) and V ∼ Γ( α = s/ 2 ,β = 1 / 2) so E U = r and E V 1 = integraldisplay ∞ 1 v 2 s/ 2 Γ( s/ 2) v s/ 2 1 e v/ 2 dv = Γ( s 2 2 ) Γ( s/ 2) 2 1 integraldisplay ∞ 2 ( s 2) / 2 Γ( s 2 2 ) v ( s 2) / 2 1 e v/ 2 dv = 2 1 s/ 2 − 1 = 1 s − 2 . 1 So E parenleftBig U/r V/s parenrightBig = s r E U E V 1 = s s 2 , for s > 2....
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 Spring '07
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 Correlation, Probability, Variance, Triangular matrix, Multivariate normal distribution, Cov, X1 Z1 =L

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