Let x x 2 bull bull bull x11 be a random sample from each

Question

Let X., X 2 , • • • , X11 be a random sample from each of the following distributions involving the parameter 0. In each case find the m.l.e. of 8 and show that it is a sufficient statistic for 8 and hence a minimal sufficient statistic.

(a) b(l , 8), where O 8 J.

(b) Poisson with mean 8 > 0.

(c) Gamma with rt = 3 and p = (} > 0.

(d) N(8, I), where -oo 8 oo.

(e) N(O, 8), where O <>8 oo.