A Level Maths Notes: S2 – The Poisson Distribution
The Poisson distribution models a situation in which
events happen at a certain rate, so many accidents in this stretch of
road per month, or so many misprints in this book per page. It is
written
,
where
is
the mean number of events in a certain time period. The Poisson
distribution has the very useful feature that it is scalable so that
if you double the time period, you double the expected number of
events. Since the Poisson distribution has only one parameter, the
expected or mean number of events,
this is a very useful feature.
The Poisson distribution is defined by
It
may be used as in the following examples.
Example: On a stretch of motorway accidents occur at a rate of 0.9 per month.
a) Show that the probability of no accidents in the next month is 0.407, to 3 significant figures.
b) Find the probability of exactly 2 accidents occuring in the next 6 month period.
c)Find the probability of at least two accidents in the next six months.
a)
to
3sf.
b)In 1 month we expect 0.9 accidents, so in 6 months we
expect 6*0.9=5.4 accidents. The distribution becomes
Using
this distribution we find
to
4sf.
c)
to
4 dp.
Sometimes two distributions are combined. The
probability of no accidents in a month is 0.407. Suppose then we need
to find the probability of having exactly 3 months in the next year
with no accidents. The probability 0.407 is fixed. The number of
months n, is 12. Of course now it is a binomial distribution,
For
a binomial distribution,
to
4dp.
This sort of thing is actually quite common, and means that every situation should be analysed carefully. It is not always the case that a single distribution should be used throughout for each question.