4.2.3 Mean of the  4.2 Applications  4.2.1 Weighted averages

4.2.2 Mean of the Poisson probability distribution function

Exercise 4.1: Consider an experiment in which a number of events (e.g., ice crystals entering the sample volume of an airborne probe) are counted under conditions where the Poisson distribution function is expected to characterize the probability. Show that the maximum-likelihood estimate of the mean number of events is equal to the number of events counted; i.e., the estimate of the population mean $\mu$ is x. (The Poisson distribution is asymmetrical, so this result is not obvious.) Also show that the resulting estimator is unbiased.