![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,](//img.homeworklib.com/images/1598fc76-8a9d-488e-80b4-6bbfc6324c2e.%2Cn~is%5C%5C%20%5Cleft%28%5Cprod_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%5Cexp%5Cleft%28-%5Cfrac%7Bz_i%5E2%7D%7B2%7D%20%5Cright%20%29%20%5Cright%20%29%5Ctimes%20%5Cleft%28%5Cprod_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%5Cexp%5Cleft%28-%5Cfrac%7Bw_i%5E2%7D%7B2%7D%20%5Cright%20%29%20%5Cright%20%29%5C%5C?x-oss-process=image/resize,w_560)
![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](//img.homeworklib.com/images/0e3f0b1b-803b-4c50-8b7d-a8869f0a1baf.%5C%5C%20Again%2C~since~S_%7BWW%7D%5Csim%20%5Cchi%5E2_%7Bn-1%7D%5C%5C%20then~S_%7BWW%7D-%5Cfrac%7BS_%7BWZ%7D%5E2%7D%7BS_%7BZZ%7D%7D%5Csim%20%5Cchi%5E2_%7Bn-2%7D~%28using~Cochran%27s~theorem%29%20%5C%5C?x-oss-process=image/resize,w_560)
![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) Us](//img.homeworklib.com/images/a1249963-4c93-4a47-b79d-878c55561891.?x-oss-process=image/resize,w_560)
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