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Poisson distribution was used in a maternity hospital
Here’s 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?
Full Answer Section
Expected days with 5 or more deliveries = P(5 or more) * 365
(ii) Greatest number of deliveries
The Poisson distribution is a discrete probability distribution, meaning it can take on only integer values. While the average is 2.74, it's possible to have higher numbers of deliveries on a given night. To estimate the greatest number of deliveries expected, we can look at the probabilities for higher values and see where they become negligible.
(iii) Reasons for deviation from Poisson distribution
There might be several reasons why the pattern of deliveries might not strictly follow a Poisson distribution:
Seasonality: Birth rates might vary slightly throughout the year due to factors like seasonal trends or cultural practices.
Day-to-day fluctuations: There might be random fluctuations in birth rates due to unforeseen events or factors.
Clustering: There might be a tendency for births to occur in clusters, which could violate the assumption of independence required for the Poisson distribution.
Changes in hospital policies or procedures: Changes in hospital policies or procedures could affect the pattern of deliveries.
By understanding these potential deviations, the hospital can refine their staffing and resource allocation strategies.
Sample Answer
Understanding the Poisson Distribution
The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. In this case, the events are births, the interval is a night (midnight to 8 AM), and the average rate is 2.74 births per night.
Answering the Questions
(i) Days with 5 or more deliveries
To find the probability of 5 or more deliveries, we can calculate the probability of 0, 1, 2, 3, and 4 deliveries and then subtract this cumulative probability from 1.
We would need to calculate P(4) using the Poisson formula, similar to the other probabilities given. Once we have P(4), we can plug all the values into the equation to find the probability of 5 or more deliveries.