Dr. Hackney STA Solutions pg 60

Dr. Hackney STA Solutions pg 60 - 4-16 Solutions Manual for...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4-16 Solutions Manual for Statistical Inference as is the variance, 2 2 Var(aX + bY ) = a2 VarX + b2 VarY + 2abCov(X, Y ) = a2 X + b2 Y + 2abX Y . To show that aX + bY is normal we have to do a bivariate transform. One possibility is U = aX + bY , V = Y , then get fU,V (u, v) and show that fU (u) is normal. We will do this in 1 the standard case. Make the indicated transformation and write x = a (u - bv), y = v and obtain 1 1/a -b/a |J| = = . 0 1 a Then fU V (u, v) = 2a 1 1-2 e - 1 2(1-2 ) 1 [ a (u-bv)] 2 -2 a (u-bv)+v 2 . Now factor the exponent to get a square in u. The result is - b2 + 2ab + a2 1 2) 2(1- a2 b2 u2 -2 + 2ab + a2 b + a b2 + 2ab + a2 uv + v 2 . 2 2 Note that this is joint bivariate normal form since U = V = 0, v = 1, u = a2 + b2 + 2ab and Cov(U , V ) E(aXY + bY 2 ) a + b = = = , U V U V a2 + b2 + ab thus (1 - 2 ) = 1 - where a 1-2 = U fU V (u, v) = a2 2 + ab + b2 (1-2 )a (1 - 2 )a = 2 = 2 + b2 + 2ab 2 + 2ab 2 a a +b u 2 2 1-2 . We can then write 1 2U V 1-2 exp - 1 2 1-2 u2 uv v2 -2 + 2 2 U U V V , which is in the exact form of a bivariate normal distribution. Thus, by part a), U is normal. 4.46 a. EX VarX EY VarY Cov(X, Y ) aX EZ1 + bX EZ2 + EcX = aX 0 + bX 0 + cX = cX a2 VarZ1 + b2 VarZ2 + VarcX = a2 + b2 X X X X aY 0 + bY 0 + cY = cY a2 VarZ1 + b2 VarZ2 + VarcY = a2 + b2 Y Y Y Y EXY - EX EY 2 2 E[(aX aY Z1 + bX bY Z2 + cX cY + aX bY Z1 Z2 + aX cY Z1 + bX aY Z2 Z1 + bX cY Z2 + cX aY Z1 + cX bY Z2 ) - cX cY ] = aX aY + bX bY , = = = = = = 2 2 since EZ1 = EZ2 = 1, and expectations of other terms are all zero. b. Simply plug the expressions for aX , bX , etc. into the equalities in a) and simplify. c. Let D = aX bY - aY bX = - 1-2 X Y and solve for Z1 and Z2 , Z1 Z2 = = bY (X-cX ) - bX (Y -cY ) D Y (X-X )+X (Y -Y ) 2(1-)X Y . = Y (X-X )+X (Y -Y ) 2(1+)X Y ...
View Full Document

Ask a homework question - tutors are online