The Poisson distribution was used in a maternity hospital to work out how many births would be expected during the night.

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 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? Note: In this real-life example, deliveries followed the Poisson distribution very closely, and the hospital was able to predict the workload accurately. Submission Instructions: Submit your assignment by 11:59 pm on Sunday. It should include at least 2 academic sources, formatted and cited in APA.

Sample Solution

Analyzing Maternity Hospital Deliveries Using the Poisson Distribution

Understanding the Problem

The Poisson distribution is a statistical probability distribution that models the number of events occurring within a fixed interval of time or space, given a known average rate of occurrence. In this case, the events are births, and the fixed interval is a night (midnight to 8 AM).

Calculating Expected Deliveries

Given:

Average number of deliveries per night (λ) = 2.74
Poisson probability formula: P(X = k) = (λ^k * e^(-λ)) / k!

Part (i): Days with 5 or more deliveries

To find the probability of 5 or more deliveries, we can calculate the complement (the probability of 0 to 4 deliveries) and subtract it from 1:

P(X ≥ 5) = 1 - P(X = 0) - P(X = 1) - P(X = 2) - P(X = 3) - P(X = 4)

Using the given probabilities and calculating P(X = 4) using the Poisson formula, we can compute P(X ≥ 5).

Part (ii): Greatest number of deliveries

The Poisson distribution is a discrete probability distribution, meaning it can take on only integer values. To find the greatest number of deliveries expected on any night, we can look for the value of X that has the highest probability. This can be done by calculating P(X = k) for increasing values of k until the probability starts to decrease.

Full Answer Section

Part (iii): Potential Deviations from Poisson Distribution

While the Poisson distribution has been found to fit the pattern of deliveries in this case, there are some potential reasons why it might not always be a perfect fit:

Seasonality: Birth rates may vary slightly throughout the year, which could affect the average rate of deliveries per night.
Clustering: There might be a tendency for deliveries to occur in clusters, such as during certain times of the day or week.
External factors: Events such as natural disasters or public health crises could temporarily affect the delivery rate.

Note: To provide more accurate calculations and insights, it would be helpful to have the complete Poisson probabilities for values of X up to 10 or 12. This would allow for a more precise analysis of the expected number of deliveries and the likelihood of extreme events.

References:

Ross, S. M. (2014). Introduction to Probability Models (10th ed.). Academic Press.
Sheldon Ross, A First Course in Probability (9th edition).

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