Please see the attached file for full problem description.

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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.

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.

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.

...frequencies for x = 0, 1, 2, 3, and 4 by using the binomial probability function. Combine categories if necessary. Observed Expected Number Frequency Frequency...

... The test Statistic used is where O is the observed frequency and E is the expected frequency. The Observed and Expected frequencies are given below. ...

... Similarly the expected frequencies for all other frequencies can be ... Where o ij is the observed frequencies and the e ij is the expected frequencies. ...

... the expected frequencies are consistently larger than the observed frequencies there are large differences between the observed and expected frequencies. ...

... row of numbers in Table 1 gives the expected frequencies under the ... in Table 1 contains the values (fo - fe)^2/fe = (Observed frequency - Expected frequency)2. ...

... is the expected frequency. The Expected frequencies are given below. Observed Expected OE (O - E)² / E % of chisq 77 110.429 -33.429 10.119 59.90 110 110.429 ...

...frequency distribution for the outcomes that you would expect from a ... to check whether there is any a difference observed and expected frequencies of the die ...

... and E is the expected frequency. The Expected frequencies are given below. observed expected OE (O - E)² / E % of chisq 78 80.500 -2.500 0.078 3.70 71 80.500 ...