Here's An Example Where The Poisson Distribution Was Used

Heres An Example Where The Poisson Distribution Was Used In A Materni

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 approximately 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 provided: P(0) = 2.74^0 e^-2.74 / 0! = 0.065; P(1) = 2.74^1 e^-2.74 / 1! = 0.177; P(2) = 2.74^2 e^-2.74 / 2! = 0.242; P(3) = 2.74^3 e^-2.74 / 3! = 0.221.

Analysis of Births Pattern Using the Poisson Distribution

This case exemplifies how the Poisson distribution predicts the likelihood of a given number of events—the number of deliveries—occurring within a fixed interval, here during the night hours. To further analyze, we address three specific questions:

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

Using the Poisson probability mass function, the probability of having 5 or more deliveries in a night is calculated by summing the probabilities for 5, 6, 7, and so on deliveries. Alternatively, it is more efficient to compute 1 minus the sum of probabilities for 0 through 4 deliveries. The probability P(X ≥ 5) = 1 - P(0) - P(1) - P(2) - P(3) - P(4).

Calculations:

• P(4) = 2.74^4 e^-2.74 / 4! ≈ 0.151
• Total probability for 0–4 deliveries = 0.065 + 0.177 + 0.242 + 0.221 + 0.151 = 0.856
• P(≥5) = 1 - 0.856 = 0.144

The expected number of nights with 5 or more deliveries is 0.144 * 365 ≈ 52.56 days, meaning approximately 53 nights per year would have five or more deliveries.

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

The Poisson distribution predicts the probability of observing a certain number of events, with the mean number of deliveries being 2.74. Although the distribution is skewed, the mode (most frequent number) for a Poisson distribution occurs near the integer part of the mean. For lam 1, it is the integer part of λ, which is 2 or 3. Thus, the most likely number of deliveries on any night is 3. To determine the greatest number of expected deliveries, we examine the tail probabilities, which become very small for higher counts. Nonetheless, the probability of, say, 7 or more deliveries is extremely low, indicating that on any given night, the maximum number of deliveries expected is quite rare, but statistically, the greater the number, the lower the probability.

Specifically, probability calculations show that counts exceeding 7 are exceedingly unlikely, but some rare nights could see higher numbers, with the probability dropping dramatically beyond 7 deliveries. Therefore, while most nights see around 2 to 3 deliveries, outlier nights might involve higher counts, but they are statistically infrequent.

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

Although in this case, the distribution closely followed the Poisson model, several factors could cause deviations in real-life scenarios. For example, hospital deliveries may not be entirely random; certain days, seasons, or demographic factors might influence birth rates, leading to clustering or periodic fluctuations not accounted for by the Poisson model. Additionally, medical or social scheduling practices, such as planned deliveries or inductions, could introduce biases, making the actual distribution deviate from the Poisson assumption of independence and constant average rate. External events, holidays, or health crises may also cause unusual patterns. Hence, in real-world data, the distribution of births may show overdispersion or underdispersion relative to the Poisson model, indicating the influence of such factors.

Conclusion

By applying the Poisson distribution in this maternity hospital setting, administrators can effectively predict workload and resource needs for night shifts. The example demonstrates the utility of the Poisson model in practical planning, provided that assumptions of independence and constant rate are reasonably met. Nonetheless, awareness of factors that could distort these assumptions is essential for accurate forecasting and effective staffing decisions.

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