Question

Only 1-6)

N(4,) "x.xx be a random sample from variance, respectively. In order to show that and let X and S be sample mean and sample 1. Let and 5 are independent, tollow the steps below. 1-1) Use the change of variable technique =nx-x,- x and show the joint pdf of ,X,,X is (n-1) n- exp f(,x) 20 2a av2 Use Jacobian for n x n variable transformation 1-2) Use the fact that N(u,a n), and show that the conditional pdf of 2 given X is n-i (n-1)s exp 2a g(x,,, ) av2 X,XX Express the conditional pdf by dividing pdf of X to joint pdf of (n-1)S 1-3) Show that the moment generating function(MGF) of for the conditional distribution of X, given X is (-1)S2 -(n-l2 E exp t -21) t<- 2 Hint: Notice that g(x,x) is a pdf. That is. (n-1)S E Cxp in a multi-integral form using the conditional pdf of 2 given X Then try to consider the integrand as another joint pdf times a constant, Then the answer will be the constant. Express and S 1-4) From the result of 1-3), how do you argue the independence of If conditional MGF does not depend on a certain statistic, then this is independent of that statistic. (n-1)S from 1-3) 1-5) Derive the MGF of Use same arqument as 1-4) (n-1)S 1-6) Find the distribution of [Hint: use Hint of problem 3 below] (n-1)S What is the MGF of Chi-square distribution and compare this with that of Mx() Mx (t) Mx (t) when X, and are independent Hint of problem 3. use

Answer 1

Kandom Xi, X2, is Xn sample from Nia") id j 2 67 n-1 n 2 -19 2 0) _ 2 Zl-xtX) 2 2-H) + 2 2/-7)(- nlx-M) d 20 )2(4-7)

2IN-X)2 n Note 2 = 1- P nx n X B&で丁 lel 1 n 2 -M)2-)+n (X- 2

1 41 M2lt) Ele") (n-)s - El F -Maltr the wt lork at hut, 1id N(M N(0g Y (Xj-MNN 2 4 M2, lt) 1-2t) u T77

1 NNn , Mylt) 2 (t-2t)"/2 New,om li) Z 2(M) )nt) unow, 2 M2 e 1Pes leoutaenors )E ( 2 + t (n-v s tn/) (n-Dst Ele S indpa CM Ms. ) M, Z2 where 22 -n n-l X-NNIO,) . . treidom -1

St MAlt) M2t) /2 (1-2 t) -2 (1-2t /-1) (1-2t) 2 U-2t) l-2 Hunt 2
PLEASE PLEASE PLEASE LEAVE A THUMBS UP!