Let X1 Xn be a random sample from a Bernoulli p distributio

Let X1, ?., Xn be a random sample from a Bernoulli (p) distribution. a) Find the MLE of p. b) Find the Cramer-Rao lower bound one variances of unbiased estimators of p. c) Find the UMVUE of p.

Solution

x1, x2....xn are bernoulli variables with two outcomes with prob for success p.

The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B(1, p) has the same meaning as X ~ Bern(p). Conversely, any binomial distribution, B(n, p), is the distribution of the sum of n Bernoulli trials, Bern(p), each with the same probability p.

Hence n=1 for all x1, x2...xn

Thus mean of x1, x2.... each = p

MLE of p = E(x1+x2+x3+...+xn)

b) Variance = pq = p(1-p) for each xi.

Var of p = VAr of x1+x2+x3+...+xn/n

= pq+pq...ntimes/n

=pq

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c) UMVUE of p = Please give meaning of UMVUE