Question

Please help me with a very detailed and a step by step approach to this transformation problem. A very self-explanatory solution will help.

a) It is about finding the joint distribution (it could be pdf, cdf, mgf, etc.) The easiest one would be preferred.

b) It is about identifying the distribution

Suppose that X1,.., xn vid N(0,1). Define k-1 Xx= x;, for k = 2, ..., n. (a) What is the joint distribution of (X2 – X2, X3 – X3,..., Xn - Xn)? What is the distribution of *(x2 – B.)? k=2

Answer 1

Solution: Ek kon ko for k=9,--- (a) The joint distribution of (X2-F2 , X3-83 - xn-n): - The joint pdf of X -Xn is, tr, flx1,x2,5. —Xa) = 4(xi) Cosme x+ are independent) = cene Exp Chef **); - <x7.com i=1,2,- xn) (91,92, Yn) Now transform (x1,x2, where yk = (k-1) XK - (XitlattXk.) 5 k(K-1) k=23, --n tan y = Nitt2 t

·cue can calily check that above transformation is orthogonal transformation and this particular transformation is called helmets transformation under this orthogonal transformation, Ê y2 = xi2 and 1 Jacobian =) . Hence, the joint pdf of Y, Yaz, n is,' = 2 2 exp (32 & f); ~ 00 zy; eo, i=1,43,-- Hence Yu ~N(0,1) independently ; K=1,2,3-n/ YK-FR = XK 12 ore = CK-12 XK -(x,+X2+ — +XK-1) - Thanks yarn co, ); K=29,– and XK XK are independent

(ie) Since yes are independent XK XK are independent - Hence, Toint distribution of (x2-F2 1 73-73) --- Xn-on) NA-160,8) where, E = biag (1/2, 2/3, 3/4,- (n-1)/n). (6) The distribution of & Kill (X K-TK) is: R=2 → 2 L. CAR- Fade com o since you hold da N(0,1); K=2,3, 2 - o-