Question

Let X1,…, Xn be a sample of iid random variable with pdf f (x; ?) = 1/(2x−?+1) on S = {?, ? + 1, ? + 2,…} with Θ = ℕ.

Determine

a) a sufficient statistic for ?.

b) F(1)(x).

c) f(1)(x).

d) E[X(1)].

Answer 1

; -0,0 +1,22 =) x 40 = X(1) TO heue.. L(240) jutte ja ut till , 02 U (X.) ; 0) = 1 if Xera ebewhere Hewe xw is a sufficient statistic by Neyman Fischer factorisation thenu fex,0).Ujeti 20, Heu f(x) = 2 1 1 2 3 + 20 th+z ootiott El. -Lethato - rxto. = 1

Ixora trinimum Free (x) & P ( xu) e al = P [ Xu yal - P[x, xx x x x greater than a home all I greater -non ... Xin x x Fridar ( 1 - 1 m ( x + o) - (Aus). х (1) = P(x1) = x) ep (Xu) ) - P(Xu) 5X). I **+0) *+ 6 012-lto) CL-6 - -16) I n(X-TFOī nate) tat chant no Stay natno ch no [(57"- 11.

b7+n 3 . +6+ ) ( 2 ) , ਦੇ 1% 2 2 2 - 1 - 1 || : 011 | [ + { 0 +) ( 4 ) +97 - 1 + : + +... ) ਧਰ3 -2 | ) N - 7 Et+2x +33 .. - % 828 +ਹਿ ॥ 8- 8 +) E|Xuy | : ( +1-1 ---