Question

Let's say (XL, Y)',.., (Xn,Yn)'(n > 2) is a random sample at Bivariate Normal Distribution ρσ102 41(ΑΓΑΝ! and r is sample correlation coefficient r- Also when Z and W, is as below, answer following questions (Question 3) (1) Show independence between (Z1, ..,Zn)' and (W, W)' (2) When (,,..,Zn)' and (Wi.,W' is independent, show Sww- Zw ~x2(n - 2), and show it is Szz ,Zn) independent with (Z1 Szw is independent, showNO,.1), and show it is 3) When (Z Z) and ( independent with (Z1, ,Zn) 2w ,Zn)' and (W,W) is independent, show independence between Sww - (4) When (Z Szz 5 and Szz Szz

Answer 1

2 where, σι The joint pdf of (Xi, Yi);i 1,2,..., n is Transform (X, X) → (ZnWǐ):-1, 2, , n The joint pdf of (Wi, Zi); i-1,2, ..., n is exp i exp -

Hence (Z1, Z2,.., Zn) and (Wi, W2,., Wn ar independent and Z, 'id N(0, 1) and W, id N(0,1); i- 1(1)n 7l 7L If Z1, Z2, ..., Zn are fired numbers then and hence nZ2~xi Swz ~ Ņ(0, 1) and hence ~ χ Szz ZZ) then Szz WZ Szz Szz Again, since Sww X-1 WZ then Sww - ~ X-2 (using Cochran's theorem)

WZ Suaare independent Szz WZ If (Z1, Z2,.., Zn) are random variables then Sww,Sw Szz of (Zi, Z^,., Z) hence 7l_ Szz ZW (3) Using (2), ~ N(0.1) VSzz Szw Zi and Cov Szz Szz Szz (Since E(Zi)0 and Zi and Ws are independent) Szw is independent with (Z1, Z2, ..., Zn) Szz Hence (4) Since Z and Ws are independent then Sww and Szz are independent zWs independent with (Z1, Za., n Szz Zn) SOs imdependent with SzZ 1 2 Again Szz ги and Szz are independent hence we say that Swwz Szz SZZ