Question

Let X1 , ..., Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of p(x; p) = p^x(1 - p)^(1-x) x = 0, 1, with E(X) = p and Var(X) = p(1 - p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1 - p)^(n-?xi) and not (1 - p)^(1-?xi)) (c) Find the maximum likelihood estimator (MLE) of p. (Remember that you already found an expression for the joint pmf in part (b).) (d) Find the Fisher information for the random sample of the parameter p. (e) What is the Cramer Rao Lower Bound for an unbiased estimator of the parameter p? (f) Is the MLE of part (c) a minimum variance unbiased estimator (MVUE) of p?

          Let X1 , ..., Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of
p(x; p) = p^x(1 - p)^(1-x) x = 0, 1,
with E(X) = p and Var(X) = p(1 - p).

(a) Find the method of moments (MOM) estimator of p.
(b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1 - p)^(n-?xi) and not (1 - p)^(1-?xi))
(c) Find the maximum likelihood estimator (MLE) of p. (Remember that you already found an expression for the joint pmf in part (b).)
(d) Find the Fisher information for the random sample of the parameter p.
(e) What is the Cramer Rao Lower Bound for an unbiased estimator of the parameter p?
(f) Is the MLE of part (c) a minimum variance unbiased estimator (MVUE) of p?

Let X1 , ..., Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of
p(x; p) = p^x(1 - p)^(1-x) x = 0, 1,
with E(X) = p and Var(X) = p(1 - p).

(a) Find the method of moments (MOM) estimator of p.
(b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1 - p)^(n-?xi) and not (1 - p)^(1-?xi))
(c) Find the maximum likelihood estimator (MLE) of p. (Remember that you already found an expression for the joint pmf in part (b).)
(d) Find the Fisher information for the random sample of the parameter p.
(e) What is the Cramer Rao Lower Bound for an unbiased estimator of the parameter p?
(f) Is the MLE of part (c) a minimum variance unbiased estimator (MVUE) of p?

Added by Lisa P.

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