This question is about a discrete probability distribution known as the Poisson distribution. Let x be a discrete random variable that can take the values 0,1,2,… A quantity x is said to be Poisson distributed if the probability P(x) of obtaining x is
P(x)=x!e−mmx,
where m is a particular number (which we will show in part (b) of this exercise is the mean value of x).
(a) Show that P(x) is a well-behaved probability distribution in the sense that
x=0∑∞P(x)=1.
(Why is this condition important?)
(b) Show that the mean value of the probability distribution is ⟨x⟩=x=0∑∞xP(x)=m.
(c) The Poisson distribution is useful for describing very rare events, which occur independently and whose average rate does not change over the period of interest. Examples include birth defects measured per year, traffic accidents at a particular junction per year, numbers of typographical errors on a page, and the number of activations of a Geiger counter per minute. The first recorded example of a Poisson distribution, the one which in fact motivated Poisson, was connected with the rare event of someone being kicked to death by a horse in the Prussian army. The number of horse-kick deaths of Prussian military personnel was recorded for each of 10 corps in each of 20 years from 1875--1894 and the following data recorded:
[table — see PDF]
Calculate the mean number of deaths per year per corps. Compare the observed frequency with a calculated frequency assuming the number of deaths per year per corps are Poisson distributed with this mean.