Question: Let X be the Poisson random variable with probability density function

\[p(x)=\frac{{{\lambda }^{x}}{{e}^{-\lambda }}}{x!}\] when x=0,1,2,3,….

Where the parameter \[\lambda \] >0. For this random variable

Consider a random sample of size n from X distribution, and let \[Y=\sum\nolimits_{i}{{{X}_{i}}}\], be the sample sum.

a. Show that Y/n is a maximum likelihood estimator for \[\lambda \].

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

c. Also, show that Y/n is an efficient estimator of \[\lambda \].

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, 20, 16, 18, 26, 27, 27, 18, 24, 22, 28, 24, 21, 17, 24

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