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Poisson distribution

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Q1: A sample of n independent observations x1, x2, . . . , xn are obtained of a random
variable having a Poisson distribution with mean ?. Show that the maximum likelihood
estimate of ? is the sample mean [see the attached file]. Show that corresponding estimator (X) is an unbiased estimator of ?, and has variance ?/n.

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 1/2 seconds duration, for 2,612 different intervals. A total of 10,126 particles were counted. The data obtained are summarised in the table attached.

It has been suggested that the number of particles emitted in an 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.

Q2: A random sample X1, X2, . . . , Xn (n > 2) of independent observations is taken from
a random variable X which has a geometric distribution with parameter p (0 < p < 1),
whose probability function is given by [see the attached file].

Information was obtained on the birth order of children in 7745 families. The records
were examined to see when a family's fi rst daughter was born. The birth order number for
the rst daughter and the data are given in the attachment.

It has been suggested that the birth order number of the fi rst daughter may be adequately modelled by a geometric distribution. Assuming this conjecture to be correct, fi nd
the maximum likelihood estimate of p (using the second result above) and estimate the
expected frequencies for the corresponding observed frequencies in the table. Comment
informally on the extent of agreement between these observed and expected frequencies.

Q3: A random sample X1, X2, . . . , Xn of independent observations is taken from a Gamma
distribution, whose density function is given by [see the attachment].

What reason would you give for supposing that the distribution is approximately normally distributed for large n?

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Solution Summary

Finding the maximum likelihood estimator from a Poisson distribution.

Solution provided by:
Chris Busu, PhD (IP)
Education
  • BSc, University of Bucharest
  • MSc, Ovidius
  • MSc, Stony Brook
  • PhD (IP), Stony Brook
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