Question

proof for distribution of (n-1)S^2/sigma^2 is the chi square distribution with n-1 degrees of freedom.

I don't understand the expansion of the square, specifically how certain terms disappeared and how a sqrt(n) appeared. Also towards the end, why does V have a degree of freedom of 1? x A detailed explanation of what happened from step 2 to step 3 would be very helpful!

THEOREM B The distribution of (n − 1)S2/02 is the chi-square distribution with n – 1 degrees of freedom. Proof We first note that Also, Ž&w=į (4) on & -w = - 1x – 5) + 7 – WF Expanding the square and using the fact that !-(X; – X) = 0, we obtain ŽX; -w)= Ž x; – x)2 + ( ) i=1 This is a relation of the form W = U + V. Since U and V are independent by Corollary A, Mw(t) = Mu(t)My(t). W and V both follow chi-square distri- butions, so Mw(t) My(t) = My(t) (1 – 2t) -n/2 (1 – 2t)-1/2 = (1 – 2t)-(n-1)/2 The last expression is the mgf of a random variable with a x72_, distribution. I

Answer 1

have ind N (4,6") Suppose, X., X. ,..., Th distribution. Then, Jo sker (*X Step 1 : X, 12,...,xu indar(n,r) . Then X J - ..., tudi are als rondom variable, that follows r(0,1) dista > *-* ~ N(0,1) vi-1071 independently. Then , we know the sum of squne of standard Normal distributo follows Chi-squre distrubution it. ( ) = 1, Step 2: Now, we break the above sum of spure, 3 (*) [ (x: + 3) + (23-)]* Ž [(6-3)** (7»)+ 2. (6-7)(z-»)] 36-35+ Ž (-) + 2 13 Mo) Ž 6. -=> )

Step 3 As x xx ... tn are random sample We know, 2 x ~ N(M," 1 from N(4,6) "E (7) FOTO von 12) = 1 / Evle) - not to le ) - 1. llevar Sum of squre of only one standard normal ] 7 (zu)** x Step 4 from we have, Ž (4:--)". = 3 (63) + (7. zu sehine, w. (..) v. (G. u. (..) Then moment generating function of w e Mult) = f(et) & (e tu eh} = Elet) & (64) M (1) Mvlt) - so. U and v are independent As w and av both are chi-squre distribution, Then. Molt). Mw (t) (1-2) I moment Mult) 2015 / funt of t is - (1-21)ka - ne a moment genarating function Hence, v. / E (x,- x) = (0-1) 5 ~ Xn, [Proved] 6