Let X be a Poisson random variable with probability density function:

p (x) = λ^x e^-λ/x!   When x = 0, 1, 2, 3, ...

Where the parameter λ > 0,

For this random variable

Consider a random sample of size n from the X distribution, and let

Y =∑_I X_i, be the sample sum.

a. Show that Y/n is a maximum likelihood estimator of λ. 

b. Show that the estimator in part (a) is unbiased and consistent.

c. Also, show that Y/n is an efficient estimator of λ.

d. In 1980, asbestos fibers on filters were counted as part of a project to develop measurement standards for asbestos concentration by the National Institute of Science and Technology.  Twenty three random samples yielded the following counts:

31, 29, 19, 18, 31,28,34,27,34,30,16,18,26,27,27,18,24,22,28,24,21,17,24

Assuming that the Poisson distribution is a plausible model in describing variability of asbestos fiber counts in filters; derive a 95% confidence interval indicating the variability in the average number in asbestos fibers.