Suppose X1,X2,...,Xn is a random sample from a binomial distribution. Recall X is the number of successes in m independent trials with probability of success at each trial is θ. We know that E[X] = mθ and V ar(X) = mθ(1 −θ). (a) Find the method of moment estimator for θ. (b) Is the estimator in (a) unbiased? (c) Is the estimator in (a) consistent? (d) Show that the estimator in (a) is also the maximum likelihood estimator.

Suppose X1,X2,...,Xn is a random sample from a binomial distribution. Recall X is the number of successes in m independent trials with probability of success at each trial is θ. We know that E[X] = mθ and V ar(X) = mθ(1 −θ). (a) Find the method of moment estimator for θ. (b) Is the estimator in (a) unbiased? (c) Is the estimator in (a) consistent? (d) Show that the estimator in (a) is also the maximum likelihood estimator.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.1: Measures Of Center

Problem 9PPS

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2. Suppose X1,X2,...,Xn is a random sample from a binomial distribution. Recall X is the number
of successes in m independent trials with probability of success at each trial is θ. We know that
E[X] = mθ and V ar(X) = mθ(1 −θ).
(a) Find the method of moment estimator for θ.
(b) Is the estimator in (a) unbiased?
(c) Is the estimator in (a) consistent?
(d) Show that the estimator in (a) is also the maximum likelihood estimator.

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