Compute Poisson probability

The Poisson distribution is well-suited for modeling the number of occurrences of an event within a fixed interval of time or space—such as arrivals, clicks, or failures.

I always begin by confirming the unit interval: Is λ expressed per hour, per day, or per page? Using the wrong unit will yield incorrect probabilities.

In the previous article, we studied the normal distribution, which appears widely in nature and social sciences—especially when modeling the sum of many independent random variables. In this article, we explore the Poisson distribution, a discrete probability distribution particularly appropriate for modeling the number of times an event occurs within a fixed time interval (or spatial region).

Definition of the Poisson Distribution

The Poisson distribution models the probability of observing k occurrences of a given event within a fixed time interval or region—provided that the average number of occurrences in that interval is known and equal to λ. The probability that exactly k events occur is given by:

To decide whether the Poisson distribution applies, first verify:

• Are events independent?
• Is the time interval (or spatial region) fixed?
• Is the average rate of occurrence stable?

$$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$

where $e$ is Euler’s number (≈ 2.71828), and $k!$ denotes the factorial of $k$.

Key Properties of the Poisson Distribution

1. Single Parameter: The Poisson distribution is fully characterized by one parameter, $\lambda$, representing the average number of events per unit time or area.
2. Memoryless Property: The underlying Poisson process is memoryless—meaning the probability of future events depends only on the current state, not on past occurrences.
3. Independence of Events: Events occur independently; the occurrence of one event does not affect the probability of another.

When reading “Common Probability Distributions: The Poisson Distribution”, start with the task, core concepts, exercises, and decision points illustrated in the accompanying figures—then return to the main text to fill in details. This approach helps you quickly assess which real-world scenarios this content applies to.

Applications

The Poisson distribution finds broad use across disciplines, including:

• Call center arrivals: Number of incoming calls received by a call center in one hour.
• Accident counts: Number of vehicle collisions at a particular intersection over a given period.
• Customer arrivals: Number of shoppers entering a supermarket during a specified time window.

Case Study: Call Center Arrivals

Suppose a call center receives an average of $\lambda = 30$ calls per hour during peak hours. What is the probability it receives exactly 25 calls in a given hour?

Using the Poisson formula:

$$P(X = 25) = \frac{30^{25} e^{-30}}{25!}$$

In Python, we can compute this using the scipy library:

import scipy.stats as stats

lambda_value = 30
k_value = 25

# Compute Poisson probability
probability = stats.poisson.pmf(k_value, lambda_value)
print(f"Probability of receiving exactly {k_value} calls: {probability:.4f}")

Output

Running the code yields the probability of receiving exactly 25 calls—helping decision-makers better understand staffing needs and service capacity during peak periods.

When reviewing “Common Probability Distributions: The Poisson Distribution”, place key concepts, procedural steps, and observable outcomes on the same page for efficient revision.

When practicing “Common Probability Distributions: The Poisson Distribution”, write input conditions, processing steps, and expected outputs together—making future review faster and more reliable.

Summary

In this article, we introduced the Poisson distribution, covering its definition, essential properties, and practical applications—with illustrative examples highlighting its real-world utility. As a tool for modeling discrete, rare, and independent events, the Poisson distribution helps quantify randomness in many operational and scientific contexts.

In the next installment, we’ll delve into the geometric distribution, further expanding our foundational knowledge of probability theory—stay tuned!