Question

$ 200, if x > 10 else 3) Let X1, X2,..., X, bei.i.d. random variables from a population with f(x;0) = 0 > 0 being unknown parameter. a) Sketch a graph of a density from this family for a fixed 0. b) Find the cumulative distribution function F(x;0) of X1. c) Show that X (1) is a minimal sufficient statistic for e.

Answer 1

_$60) 2.70 a for 0-1 f(x) = 2 x 71 + 23+ 9 6) FLH,0), do joo dx = 20 ( of(,) = 207(X>0) 104,0 16ve) +(4,0) *O1140) TY - 0 f (2,0) 3 I (X>0) Fly,o) - Tx I (150) is independent of o when X = Yu Hence is the minimal sufficient statistic for o.

d) We have the distribution tunition of toy is. P(X., EY) = 1- P(x, y, X , 74, ... X, Y) = 1-{P ( x, 3, 4)] = 1- (1 - P(x, y)] Then 1-GM density of tw is the 2n (2n+1) = - 0 xy x-2n x-2n an = 2 no y ent1 a y 30. 2 ( ano 21+1 ) 9 70 otherwise . EO) Byong day at y s dy 19 - 2n+ 2n - ano 2n zno 1-1 E(X) = 2 non ano (1-2n)

e) fixo) - 20,*70 nan ie, to Here range depends on parameter, and is sufficient for o. {f(x,0) = 2 0 I ( x, 70)} by TMT factorization theorem . Hence since range depends on parameter the ordered statistic X is complete sufficient for o {if you need complete prout by definition please comment I will do it and reupload}, We have a result that it any tunition of complete sufficient statistic is unbiased then that tunition is UMVUE from part (d) we get Elxw ) = 200 zno (1-20) > F(- 10 %) = 0 » UMVUE of o= (1-2n) X 2n.