Question

$X_{1,}X_{2,}X_{3}$.........,$X_{n}$, $N(\mu ,\sigma ^2)$ random sampling of the normal distribution of the unit n, and and ile _{$S^{2}_{}n^{_{}}$ -1}  Let the sample mean and sample variance be respectively.

a)  $\bar{X}\sim N (\mu ,\sigma ^2/n)$

b)   ile $S^2_{n}_{}$ _-1  it is independent.

c)  $(n-1).S^2_{n-1}/\sigma ^2\sim \chi ^2_{(n-1)}$

d)  $Var(S^2_{n-1)}=$$2\sigma ^4/(n-1)$

What is the proof ??

Answer 1

Solution Lotu) = exp 2 i Let us Tromsform by means of to the variables all = a12= Oo. ain It can be easily the The joint pdf of Xiscongan is given by fraue lover (Xi–412 ],,-46.76 do Yig (1=1,2000n) a linear orthogonal transformation Y = AX Let us choose in particular I kn 9,= tan (xit tan)=rł seen thank the above choice of aniq a,29ooo ain satisfies condition of since the Troms formation is orthogonal, we have Eyiz ={n xi² = {"(ai_222 +nn². ia {y; ² = [ (i-1) 2 + gp² [ yiz" xi-)2 -- (**) 1=2 An (xi-u? = {n (ainīt sł -42 = 2(xi-x3²7 ncł-M) 2 = 4; 2 tr(8-4)? orthogonality a viz, n Σ' ja aij?=1 121 n 1-1 Also, -1 121 n 122 The Jacobiam n of Transformation tion J=Ili Thus the joint density function of Xuxn Troms forms to ag (y, ... yn) = ( toren) exp [ hoz y;²+ ne (ă-423] *131 122 Ics banned with Camscanner

Seat (1-4)23 Ix 202 ke { ace i=2 pl n 2 En g 141... Yn) = [terco Exp -(olund n-1 [laza) exp i-2 202 Thus it and 29; 2 = {(xi-x)² = ns². independently distributed which meom (n-1)2 = {(xi-72 and X are independenty distributed and XON(MIG) and yo (1=2, ..n) are independent N10102) hence Z Yi? iz282 being the sum of squares of child independent standard variate is distributed as x2 variate (n-1) df. a) Xan(4,02/n) Roof 6) K 6 shol is independent Boof () Mish ham d) voord 5 ) var Z vas (srl) = 204 M-1752 o2 и32. no in normal with so, x²ent) N bor 02 varmasta) = 2.16) in-u var 15 hp)=254 ch) cs