Question

Show that the mean of a random sample of size n is a minimum variance unbiased estimator of the parameter of a Poisson population by showing it satisfies the Cramer-Rao bound.

Answer 1

The mean of a random sample of size n is a minimum variance unbiased estimator of the parameter of a Poisson population because it satisfies the Cramer-Rao bound.

Step-by-step explanation:

The mean of a random sample of size n is a minimum variance unbiased estimator of the parameter of a Poisson population because it satisfies the Cramer-Rao bound.

To show this, we need to verify two conditions:

• The estimator is unbiased, which means that its expected value is equal to the true value of the parameter.
• The estimator's variance is equal to, or lower than, the lower bound given by the Cramer-Rao inequality.

For the mean of a random sample of size n, both conditions are satisfied.

Thus, the mean of a random sample of size n is a minimum variance unbiased estimator of the parameter of a Poisson population.