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# 1

Let X1,..., X,, be a random sample from the Bernoulli distribution, say P[ X = 1] = 0 =1 - P[X = 0]. (a) Find the Cramer-Rao lower bound for the variance of unbiased estimators of 0(1 - 0). (b) Find the UMVUE of 0(1 - 0) if such exists.Let X1, ..., An be a random sample from the binomial distribution (xp(1 -P)m-*, x= 0, 1, ..., m, where m is known and 0 Sp $1. (a) Estimate p by the method of moments and the method of maximum likelihood. (b) Is there an UMVUE of p? If so, find it