2ng 2 d) Show that the density of X ( is given by fx,, (y:0) = yenIT , ify 2 0 and hence find EX (1)- Hint : Use P (X(1) y, X2 > y, . . ., Xn > y) = 1 - [P(X1 > y)]" and b) to find the cumulative distribution function of Xy, then differentiate it. c) Show that X(1) is complete . Hence find the UMVUE of 0. 4) a ) Suppose a sample X = (X1, X2, ..., X,) of i.i.d. random variables with a common density f(x, 0) = 01(0-1)(0 0 is unknown. i) Find a minimal sufficient statistic T for 0. ii) Show that W = - log X, is an unbiased estimator of T (0) = . Hint: You may use the density transformation formula fw (y) = fx (x)| | to identify the distri- bution of W first - iii ) Find the UMVUE of T(0) = y. Justify your answer. (*) iv) Find the UMVUE of T (0) = 0. Justify your answer. (*) 5) For a sample of n i.i.d. random variables X1 , X2, ..., X, from a distribution with a finite variance o': i) Show that the estimator S? = _ EF, (X, - X)? is unbiased for a? = Var (X;).j = 1, 2, . .. . 74. ii) The Rao -Blackwellization generally decreases (or at least does not increase ) the variance of an unbiased estimator without violating the unbiasedness property. Now assume that your sample is from a Bernoulli (0) distribution . Rao-Blackwellize $2 from the first principles (i.c., by conditioning on the complete and sufficient statistic T = 2" ,X;). Show that the resulting estimator can be written in the form X (1 -X).". Hint: For fixed j = 1, 2, . . ., n you may want to determine the probability P (X ; = k[T = (), k = 0, 1 first . iii) Does X(1 - X)_" have a smaller or same variance as S' for your Bernoulli sample? Justify your