Inference statistics

Let X11 X2. X" be the survival times of a random sample of rr identical electronic compo- nents. and suppose that the survival time of a component has a distribution with probability density function m\""]"' \" 5 x -=: oo. fillt'} = I} otherwise. where )3 3 (land a: r:- it. {a} Write down the joint density function of XI. X2, X... [2) {b} Prove or disprove that \"Xi, 21'.) is a jointly sufficient statistic for (a, }. [3) '2] .I'=l to} Prove or disprove that \"Xi, x.) is a jointly minimal sufficient statistic for (a. ,6}. '2] 1'2] {3) {d} Is (1] 13-. 2 X5) a complete sufrmient statistic for (a, m. (a) 1'2] l7=l {e} Prove or disprove that f {1|a, ,8] belongs to the exponential family of probability density functions. That is. show that f ma, ,3} can or can not be expressed as .I' f{x|} = Ir{x)c{}exp [Zim- {inn-{1}] for: E x C R and H E G C Hi. i:] where th) :5 i} does not depend on 6'. out} :2: f} and mull}, mafia), mm} are real- valued functions of the parameter E [they cannot depend on x). [6} {f} STEM revision. Find maximum likelihood estimators of o and ,8. {7} 3 to} snare: revision. Find the Crammer-Hoe lower bound of the variance of an unbiased estimator of: {i} a; and {ii} ,3. {I}