Question

The water quality of a river was investigated by taking 15 water samples and recording the counts of a particular bacteria for each sample, where ?? is the number of bacteria in sample ?? and the ??’s are assumed to be independent and follow a Poisson ( ?oi(?) ) distribution. The data are as follows: {27, 24, 25, 21, 30, 22, 20, 22, 29, 19, 19, 22, 23, 36, 19}

1. Find a point estimate for the population mean number of bacteria in the water using maximum likelihood estimation.

2. Use R to plot the log-likelihood function with the maximum likelihood estimate highlighted.

3. Show that the maximum likelihood estimator of ? is a consistent estimator.

4. Show that the maximum likelihood estimator of ? achieves the Cramer-Rao lower bound.

5. Derive the asymptotic distribution of ? ^, the MLE of ?.

6. Find an approximate 95% confidence interval for ? using the Wald method

Answer 1