Question

25 Let X1, X, be a random sample from f(x; 6) =f(x; M, O) = 44.02(x). Define (0) by Moon 04.62 (x) dx = a(a is fixed). Recall what the UMVUE of 7(0) is. Find a 100 percent confidence interval for (O). (If you cannot find an exact 1007 per- cent confidence interval, find an approximate one).

Answer 1

25 ܘܘ Solution : Given information ºs * Let us US consider XiX2,- Xn be a random Sample from +(4; 6) =+(2;4.60) -% (2) عسير * Now , define e (0) by o (x) da X (a is fixed) TO) '",o2 X~Quare x-ll Р e 1 p(x >70) (2 T (0) - " 1 - P ( 1(0)) -P(2 (0)-4) => x-ll c(0) < 2-1 -> Р de-M < I (0) - 1 - 2 x-u N (0,1)

* Let us take § (1) be the CDF of CDF of N(0,1) Р V T (0) - u - 1-a => - ? (0) - 11 (1-2) => C (6) - (1–2)) > (1-2) tu (14,2%) > Xorll ~N (0,1) V => (Yu-heye ~ x 1 n Σ i=1 (x:- lej? Xin 2 22 we know that A130, se (x, xJ² n Σ (n-1) = 2 ?

3 = (n-1) (n-1). S x (2-1) 6 * For y~x n E (78) (CS+ E [(a)23 2 2 2 n-) + 2 no r 2 Intl فلا e forbunying on xr) n 2 (n-1) 52) x í 2 E re E (Jn-1.595Y8 veira - > = xa n-1 st -1.5) r-> rer (2) os unbiased for o,

(4) Already we know that I E Xo 98 unbiased for lo : ātnis (12) r(n) (1-2) is unbiased 2 - for Mt of (1-2) = 2(0) (where 0 = 4 N(D, 6%) (7,5) is complete and Sifficient Therefore by using Lehman Scheffe theorem, Wnts I (0-1) (1-x) S UMVUE of 700 Te rom X t lx here st SWT & xi and n S (x,-5) n-l