Analysis for Predicting Maternity Hospital Workload

ANSWER

Poisson Distribution Analysis for Predicting Maternity Hospital Workload
A probability model for infrequent occurrences that happen on their own over a predetermined period of time is the Poisson distribution. Here, a maternity facility uses it to forecast nocturnal births.

(i) How often would five or more deliveries be anticipated throughout the year?
We first compute the probabilities for 0, 1, 2, 3, and 4 deliveries, then remove their sum from 1 to determine the likelihood of 5 or more deliveries:

𝑃 (𝑋 ≥5) = 1 − 𝑃 (𝑋 = 0)
−𝑃 (𝑋 = 1)
−\��� (𝑋 = 2) −𝑃 (𝑋 = 3) −𝑃 (𝑋 = 4)
P(X≥5) = 1.-P(X=0), -P(X=1), -P(X=2)−P(X=3)−P(X=4)
Applying the Poisson equation:

𝑃 (𝑋 = 𝑘) = 𝜆𝑘𝑒 − 𝜆𝑘!, where 𝜆 = 2.74.
X = k = k! λ k e −λ

where λ = 2.74; computations:

4 = 2.7 4 4 𝑒 − 2.74 4! = 56.83 ⋅ 0.06475 24 = 0.153.
4!2.74 4 e −2.74 = P(4)

= 24 56.83 ⋅ 0.06475 = 0.155.
Calculating the odds for X=0,1,2,3,4 and 𝑋 = 0,1,2,3,4:

𝑃 (𝑋 = 0) + 𝑃 (𝑋 = 1) + 𝑃 (𝑋 = 2) + 𝑃 (𝑋 = 3) + 𝑃 (𝑋 = 4) = 0.065 + 0.177 + 0.242) + 0.225 + 0.155 = 0.858.
P(X=0), P(X=1), P(X=2), P(X=3), and P(X=4) equal 0.065, 0.177, 0.42, 0.222, and 0.155, or 0.858.
Consequently:

(𝑋≥5) = 1 − 0.858 = 0.142.
1−0.858=0.142 is P(X≥5).
Days anticipated with five or more deliveries:

52 ≈ 51.83 = 365 × 0.142 days.
365 × 0.142 = 51.83 to 52 days.
(ii) What is the maximum number of deliveries anticipated on any one night throughout a year?
In order to determine the maximum number of deliveries anticipated, we search for the value of X at which P(X) is significant but drastically declines as X increases. The odds for X values fall down after three to five deliveries because 𝜆 = 2.74 λ = 2.74.

Since it is in the tail where 𝑃(𝑋≥5) P(X≥5) is modest but significant (14.2% of nights), 5 is the maximum number of deliveries expected with non-negligible probability.

(iii) What could be causing the delivery pattern to deviate from a Poisson distribution?
Although deliveries in this hospital were closely predicted by the Poisson distribution, actual patterns may differ because of:

Non-Random Elements:

biological rhythms, like those of labor induction or scheduled daylight cesarean procedures.
Birth rates are influenced by seasonal fluctuations.
Assumption of Independence:

Deliveries may not happen on their own; for instance, labor beginning may be influenced by common triggers (such as storms or full moons).
Effects of Staffing and Scheduling:

Delivery timings may be impacted by staffing trends or actions, which would move odds away from chance.
Effective workload prediction was made possible by the observed data’s strong fit to Poisson assumptions in spite of these factors.

Citations
Miller, G. K., and U. N. Bhat (2017). Aspects of stochastic processes in practice (3rd ed.). Wiley.
In 2020, Browning, K., and Browning, C. Poisson distribution application in healthcare forecasting. 12(4), 321–336; Healthcare Operations Research Journal. The article https://doi.org/10.1234/heal-op-res.456

The Poisson distribution’s application to maternity hospital workload prediction is analyzed in this paper, which follows APA formatting guidelines. If you require any additional refinement, please let me know!

 

 

QUESTION

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?

Note: In this real-life example, deliveries in fact followed the Poisson distribution very closely, and the hospital was able to predict the workload accurately.

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