I wonder if my solution is right and how to prove the completeness

11. Let X1, ..., Xn be a random sample from the Pareto distribution with pdf f(x|0) = J 2ex-(0+1), x2 2, 0, otherwise. (a) Show that T = EM, log X, is complete and sufficient for 0. (b) Find the UMVUE of 0. ? ? a ) 1 prove sufficiency f(x 10 = 20 . 0 " [ T x ]-(at) = 210 . on fe [is ( x1 ] - ( #+ 1 ) = 2 40 . ". e- C+DT - g ( Tc / 10) the 9 (T 10 ) = 210. 0 ". e - cotUT By Factorization Theorem, T= [milog Xi is sufficient for 2 prove completeness E (9 ( T) = 12 g( Inkgx ) . 20 0 . x 1+ d x = 20 . 0 5 9 ( [m ( ( x 1 ) x- 18- da = 0 4 0 ED 30k 10 ) = S2 80 [ g ( [ - 6 ( XD ) 2 - ( # + ) ] da = 0 = Sig (City (xol . 3 x - Com) b E [ [ FIX ; ] = GilEClogx] =n E[/pgx:] =n S2 log by 20. 0. x - 10+ doc = n- 20. 0 . S2 10 2 2x 2-0 dx = - n. 20 -) 1060. x- 0120 - 52 2. xo dx] = -n 20 co- [ ] ] 2 ] = - 820 (0-2#) = n Let I" = [ log xi "; be a function of T & ECT ( ] = n - 2=0 3 TH is the UNIQUE of 142