1. Let X., X2 , • • • , X11 be a random sample from a Poisson distribution with mean 0. Find the unbiased minimum variance estimator of 02.

2. We consider a random sample X1 , X2 , • • • , X11 from a distribution with p.d.f./(x; ()) = (l/0) exp ( -x/0), 0 <>x <>oo, zero elsewhere, where O 0. Possibly, in a life testing situation, however, we only observe the first r order statistics, Y1 <>Y2 <>··· Yr-

(a) Record the joint p.d.f. of these order statistics and denote it by L(O).

(b) Under these conditions, find the m.l.e., 0, by maximizing L(O).

(c) Find the m.g.f. and p.d.f. of 0. .

(d) With a slight extension of the definition of sufficiency, is O a sufficient statistic?

(e) Find the unbiased minimum variance estimator for 0.

(f) Show that Y1 /8 and tJ are independent.