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Poisson distribution was used in a maternity hospital to work out

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

Outbreaks: Outbreaks of infectious diseases or other unforeseen events could significantly increase the number of deliveries on certain days, deviating from the expected random pattern.
Staffing levels: Changes in staffing levels or availability of beds in the hospital could indirectly influence the number of deliveries accepted on a given night, affecting the randomness of the process.
Day of the week effects: There might be subtle variations in the number of deliveries on different days of the week.

Important Note: This analysis provides a simplified model. In real-world scenarios, more sophisticated statistical methods might be necessary to accurately predict the number of deliveries and ensure adequate staffing levels.

Sample Answer

i) On how many days in the year would 5 or more deliveries be expected?

To find the probability of 5 or more deliveries, we need to calculate the probability of 0, 1, 2, 3, and 4 deliveries and then subtract this from 1.
We already have the probabilities for 0, 1, 2, and 3 deliveries.
We need to calculate the probability of 4 deliveries using the Poisson formula:
- P(4) = (2.74^4 * e^-2.74) / 4!
- P(4) ≈ 0.126
Probability of 5 or more deliveries = 1 - (P(0) + P(1) + P(2) + P(3) + P(4))
- = 1 - (0.065 + 0.177 + 0.242 + 0.221 + 0.126)
- ≈ 0.169
Expected number of days with 5 or more deliveries = 365 days/year * 0.169 ≈ 61.54 days/year

Therefore, on approximately 62 days in a year, 5 or more deliveries would be expected.

ii) Over the course of one year; what is the greatest number of deliveries expected on any night?

While the Poisson distribution theoretically has no upper limit, the probability of extremely high numbers of deliveries decreases rapidly.
Given the average of 2.74 deliveries per night, it's highly unlikely to have an extremely large number of deliveries on any single night.
However, it's important to consider that the Poisson distribution provides probabilities, and there's always a small but non-zero probability of even very rare events occurring.

iii) Why might the pattern of deliveries not follow a Poisson distribution?

Seasonality: Birth rates may exhibit seasonal variations, which would violate the assumption of a constant average rate in the Poisson distribution.
Scheduled procedures: An increase in scheduled cesarean sections or inductions during specific times could disrupt the random pattern expected in the Poisson distribution.