# Observed and expected frequencies

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? A sample of n independent observations are obtained of a random variable having a Poisson distribution with mean . Show that the maximum likelihood estimate of is he sample mean show that the corresponding estimator is an unbiased estimator of , and has variance .

The scientists Rutherford and Geiger reported an experiment in which they counted the number of alpha particles emitted from a radioactive source during intervals of 7.5 seconds duration, for 2612 different intervals. A total of 10126 particles were counted. The data obtained are summarised in the below.

Number 0 1 2 3 4 5 6 7 8 9 10 11 12 >12 Total

Frequency 57 203 383 525 532 408 273 139 49 27 10 4 2 0 2612

It has been suggested that the number of particles emitted in a n interval may be adequately modelled by a Poisson distribution. Assuming this conjecture to be correct, find the maximum likelihood estimate of the mean of this distribution, and use this to estimate the expected frequencies corresponding to the observed frequencies given in the table. Comment informally on the extent of agreement between these observed and expected frequencies.

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#### Solution Summary

This shows how to find the maximum likelihood estimate of the mean of a Poisson distribution, and estimate the expected frequencies corresponding to observed frequencies.