When Do Random Events Follow a Poisson Distribution? A Practical Guide

Poisson Distribution

The Poisson distribution is a discrete probability distribution commonly used in statistics to describe the number of random events occurring within a fixed interval of time or space.

Probability Mass Function

λ represents the average rate of occurrence per unit interval.

k! denotes the factorial of k, where k is a non‑negative integer.

Conditions for Poisson Applicability

The event has a low probability of occurring in a very short interval.

Occurrences in each interval are independent of each other.

The probability of occurrence remains constant over time.

Example: At a bus stop, on average two buses arrive every five minutes. What is the probability that exactly five buses arrive in a five‑minute period?

Solution: The arrival of buses satisfies the Poisson assumptions, so we apply the formula: P(X=k=5) = (2^5/5!)*e^(-2) The resulting probability is approximately 0.0361.

Other phenomena that follow a Poisson distribution include the number of alpha particles emitted by a radioactive source, the number of phone calls received by a switchboard, aircraft landings at an airport, customers served by a salesperson, and thread breaks in a spinning machine.

Poisson Cumulative Distribution Curve

Curve Characteristics:

When k is less than λ, each increase in k accelerates the cumulative probability.

When k exceeds λ, the cumulative probability decelerates more rapidly.

Why Do Most Real‑Life Situations Follow a Poisson Distribution?

If an event occurs randomly over time and meets the following three criteria, it is modeled as a Poisson process:

Dividing the time interval into infinitesimally small sub‑intervals, the probability of a single occurrence in a sub‑interval is proportional to its length.

The probability of two or more occurrences within a sub‑interval is essentially zero.

Occurrences in different sub‑intervals are independent.

Hospital example: If a day is divided into hours, minutes, or seconds, the probability of a patient arriving in an extremely short interval approaches zero (condition 1). The chance of two patients arriving simultaneously in such a tiny interval is virtually impossible (condition 2). If arrivals are independent, the process satisfies condition 3; otherwise, it does not follow a Poisson distribution. In short, when events are independent, they are likely to be described by a Poisson distribution. —Excerpt from Zhihu: 楚小鱼