The Poisson formula
Thankfully, being killed by a horse-kick is a rare event. The most suitable
theoretical technique for modelling how often rare events occur is to use
something called the Poisson distribution. With this technique, could Bortkiewicz
have predicted the results without visiting the stables? The theoretical Poisson
distribution says that the probability that the number of deaths (which we’ll call
X) has the value x is given by the Poisson formula, where e is the special number
discussed earlier that’s associated with growth (see page 24) and the exclamation
mark means the factorial, the number multiplied by all the other whole numbers
between it and 1 (see page 26). The Greek letter lambda, written λ, is the
average number of deaths. We need to find this average over our 200 corps-
years so we multiply 0 deaths by 109 corps-years (giving 0), 1 death by 65
corps-years (giving 65), 2 deaths by 22 corps-years (giving 44), 3 deaths by 3
corps-years (giving 9) and 4 deaths by 1 corps-year (giving 4) and then we add
all of these together (giving 122) and divide by 200. So our average number of
deaths per corps-year is 122/200 = 0.61.
The theoretical probabilities (which we’ll call p) can be found by substituting
the values r = 0, 1, 2, 3 and 4 into the Poisson formula. The results are:
It looks as though the theoretical distribution is a good fit for the experimental
data gathered by Bortkiewicz.
First numbers
If we analyse the last digits of telephone numbers in a column of the
telephone directory we would expect to find 0, 1, 2,... , 9 to be uniformly
distributed. They appear at random and any number has an equal chance of
turning up. In 1938 the electrical engineer Frank Benford found that this was not
true for the first digits of some sets of data. In fact he rediscovered a law first
observed by the astronomer Simon Newcomb in 1881.