b) 1 2 , ,..., X X X n is a random sample from the uniform distribution between and 1 i.e 1 f x( ) (1 ) , x 1 where is an unknown parameter. Denote the sample mean by X . (i) Show that the method of moments estimator, , of is 2 1 X . (3mks) (ii) Show that is unbiased and find its variance. (4mks) (iii) Let Y X X X min{ , ,..., } 1 2 1n . By finding P Y y ( ) , show that the pdf of Y is 1 (1 ) ( ) , 1 (1 ) n n n y f y y (4mks) c) (i) Define what is meant by a sufficient statistic (2mks) Let 1 2 , ,..., X X X n be a random sample from 1 ( ; ) (1 ) , 0,1 x x f x x , 0 1 (ii) Show that T X i is a sufficient statistic for . (3mks) d) The random variable X has an exponential distribution f x( ; ) x e , x 0 ; 0 (i) If 1 2 , ,..., X X X n is a random sample from this distribution, derive maximum likelihood estimator of . (3mks) (ii) Derive expectation of the distribution (2mks) (iii) Show that is an unbiased estimator of .