Heres an example where the Poisson distribution was used in a maternity hospital to work out how many births would be expected during the night.
The hospital had 3000 deliveries each year, so if these happened randomly around the clock 1000 deliveries would be expected between the hours of midnight and 8.00 a.m. This is the time when many staff is off duty and it is important to ensure that there will be enough people to cope with the workload on any particular night.
The average number of deliveries per night is 1000/365, which is 2.74. From this average rate, the probability of delivering 0, 1, 2, etc babies each night can be calculated using the Poisson distribution. Some probabilities are:
P(0) 2.740 e-2.74 / 0! = 0.065
P(1) 2.741 e-2.74 / 1! = 0.177
P(2) 2.742 e-2.74 / 2! = 0.242
P(3) 2.743 e-2.74 / 3! = 0.221
(i) On how many days in the year would 5 or more deliveries be expected?
(ii) Over the course of one year; what is the greatest number of deliveries expected on any night?
(iii) Why might the pattern of deliveries not follow a Poisson distribution?