Question

1. Let X1, ..., Xn be a random sample of size n from a normal distribution, X; ~ N(M, 02), and define U = 21-1 X; and W = 2-1 X?. (a) Find a statistic that is a function of U and W and unbiased for the parameter 0 = 2u – 502. (b) Find a statistic that is unbiased for o? + up. (c) Let c be a constant, and define Yi = 1 if Xi < c and zero otherwise. Find a statistic that is a function of Y1, ...,Yn and also unbiased for Fx(c) = 0 (C ).

Answer 1

i Here ,1, 2, n, Here 2 AVarl) - 0 El) i1,2, n Mat if Know We a wandom X,X2 Xn - with Size Sample of and ElX)A Var(xi) 2,.,n then unbiased an e stimator ot uand unbiased i s an eshimator

where 2X 2 ETX) M and 2 Els) hat Note X - 2 Z0i-X) n-l 1 2xn 2

2 n n-l Let T 2 2 X- 55 T 2. U 5. W - Now, E T) E 2X-55 2E()-5E(s) LO

2 ELT) {w- T: 2 U n-l i s unbiased eshimator 0 e-2 u-5 2 Vanli)= Here I,2 NoW Vanlxi) = σ r. E E (Xi) 2 Elx) 2 UV ElX)0

Let, V 2 NoW Elv) 2 ET Elx) 2 (

t E V) W unbiased V: for 10 M Here Y: -(, if Xie 7. 0.w. f Xisc .f xic PlY:- 1) PlXiC) .

PlY: -0) Pl Now, ELY:) 2y:xPlY:y) y: 1x Pl ) + 0x PliC) PLEC) El Y:) PXi

NuM plxie) ( C-M (%G Let Zi Xi-M Zi~NIO,1) Then edf of he N(O,1) ( ElY:) izl 2

NoW El 9) ( MELY:) Y: n is an unbiased estimator f F,IC) X