• Understand the basic idea of the Poisson distribution and the situations where it applies
• Compute probabilities in simple Poisson distribution problems
• Model rare real-life events with the Poisson distribution
Deeper study
• Understand and use the probability mass function of the Poisson distribution
• Understand why the mean and variance of a Poisson distribution are equal
• Apply the Poisson distribution in fields such as traffic, telecommunications, and quality control
Real-world practice
• Understand and use the properties of the Poisson process and its extended models
• Apply the relationship between the Poisson and gamma distributions in Bayesian methods
• Design Poisson models for complex real-world problems and use them in decision-making
At a glance
The Poisson distribution is a discrete probability distribution for the number of rare, independent events that occur in a fixed interval of time or space at a constant average rate λ.
Its probability mass function is P(X=k)=λ^k·e^(−λ)/k!, and its mean and variance are both equal to λ. For example, with an average of 4 events per hour (λ=4), the probability of exactly 3 events is about 0.195.
The binomial distribution B(n,p) converges to the Poisson distribution as n→∞, p→0 with np→λ, and for λ≥10 the Poisson can be approximated by the normal distribution N(λ, λ) (with a continuity correction).
In a Poisson process the waiting times between events follow an exponential distribution; the Poisson model is a poor fit for overdispersed data (variance greater than the mean) and for zero-inflated data.
The Poisson distribution is a discrete probability distribution, so you should first understand what a probability distribution is
Binomial Distribution
The Poisson distribution is derived as a limiting case of the binomial distribution, so it helps to learn the binomial first
Basic: What Is the Poisson Distribution?
Difficulty 3/5
About 25 min
The Poisson distribution models how many times a rare event occurs within a fixed interval of time or space. You use it when events happen independently of one another at a constant average rate.
Picture a hospital emergency room. If, on average, 3 emergency patients arrive per hour, you can compute the probability that exactly 2 arrive in a given hour, or exactly 5. Each arrival is independent of the others, and the average arrival rate stays the same. Modeling events like this — rare, but occurring at a steady rate — is exactly what the Poisson distribution does.
Key points
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Rare events occurring within a fixed interval of time or space
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Events occur independently of one another
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The average rate of occurrence (λ) is constant
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The probability of two events occurring at exactly the same instant is 0
A simple example
If a phone rings 4 times per hour on average (λ=4), the probability that it rings exactly 3 times in a given hour is P(X=3) = e⁻⁴ × 4³ ÷ 3! = 0.0183 × 64 ÷ 6 ≈ 0.195
Check your understanding
Answer:
In a Poisson distribution the mean and the variance are equal. Both are equal to the parameter λ.
Answer:
Use it to model counts of rare, independent events — for example the number of traffic accidents, radioactive decays, or customer arrivals in a given period.
The probability mass function of the Poisson distribution Loading... is Loading..., where Loading... is the average rate of occurrence and Loading... is Euler's number. For small Loading... the distribution is strongly right-skewed; as Loading... grows it becomes more symmetric, and once Loading... it can be approximated by a normal distribution.
Mathematical formula
Loading... is the average rate of occurrence, Loading... is the actual number of occurrences Loading..., and Loading... is Euler's number, approximately Loading.... Loading... is the factorial of Loading....
Learn through examples
Example 1: Traffic Safety Analysis
An intersection averages 2 traffic accidents per month
The probability of an accident-free month is 13.5%, and the probability of 3 or more accidents is 32.3%.
Even with an average of 2 accidents per month, a month with 3 or more accidents happens roughly one month in three (32.3%). That is why a cluster of accidents in one month should not lead you straight to the conclusion that "the intersection has become dangerous" — it is natural variation even when the rate λ has not changed. By the same logic, an accident-free month (13.5%) does not prove that a safety measure worked.
Example 2: Call Center Operations
A call center receives 15 calls per hour on average
Poisson(15): probability of 20 or more calls in an hour
The probability of receiving 20 or more calls in an hour is about 12.3%.
The question asks about "20 or more", yet the calculation uses 19.5 — that is the continuity correction. When you approximate a discrete distribution (Poisson) with a continuous one (normal), you shift the boundary by 0.5. Without the correction, (20−15)/√15 ≈ 1.29 gives a probability of about 9.8%, an underestimate. Even though λ ≥ 10 makes the approximation itself reasonable, whether or not you apply the continuity correction changes the answer.
The probability of a defect-free square meter is 44.9%, and the probability of 2 or more defects is 19.2%.
Even though the average is below one defect (0.8), the chance of a defect-free piece is only 44.9% — more than half of the pieces have at least one defect. Also, if you inspect 2 m² instead of 1 m², λ becomes 1.6 and the defect-free probability shrinks multiplicatively to e⁻¹·⁶ = (0.449)² ≈ 20.2%. In a Poisson model, whenever you change the unit of observation (area or time), λ must scale in proportion.
Common mistakes
Mistake:
Applying the Poisson distribution when events are not independent
Why is it wrong?
The Poisson distribution requires events to occur independently of one another. It does not apply to chain reactions or to events that depend on each other.
Correct approach:
Check that the events are independent, and use the Poisson distribution only when independence holds.
Mistake:
Ignoring a rate that changes over time
Why is it wrong?
The Poisson distribution assumes a constant rate. If the rate differs by time of day, for example, a single Poisson model overlooks that.
Correct approach:
When the rate changes, split the time into intervals and use a different λ for each interval.
Mistake:
Skipping the continuity correction when using the normal approximation for large λ
Why is it wrong?
For λ≥10 you can approximate the Poisson with a normal distribution, but the result is accurate only with a continuity correction (±0.5).
Correct approach:
Always apply the continuity correction when approximating a Poisson distribution with a normal distribution.
Case study: Power Outage Forecasting for an Electric Utility
Background
A major electric utility in South Korea faces a rising risk of outages as climate change brings more extreme weather and electricity demand surges. To respond efficiently and position resources in advance, the company urgently needs a data-driven outage forecasting system.
The problem
The existing experience-based approach struggles to cope with unpredictable outages. The utility needs to quantify outage patterns by season and weather conditions and build a proactive response system on top of that analysis.
The data
Five years of outage data (monthly aggregates):
• Normal periods (spring/fall): mean 1.8 outages per month, standard deviation 1.3
• Summer (June–August): mean 3.2 per month, up to 12 during heat waves
• Winter (December–February): mean 2.1 per month, up to 8 during cold snaps
• Typhoon season (July–September): mean 8.5 per month, up to 25 under a direct typhoon hit
A forecasting system built on the Poisson distribution let the utility quantify outage risk and deploy resources efficiently. Bayesian updating in particular allowed it to adapt to changing conditions in real time, improving both the reliability and the economics of the power supply.
References
Poisson, S.-D. (1837), Recherches sur la probabilité des jugements en matière criminelle et en matière civile, Paris: Bachelier — the original work in which the Poisson distribution first appears
von Bortkiewicz, L. (1898), Das Gesetz der kleinen Zahlen, Leipzig: B. G. Teubner — the classic first serious application of the Poisson distribution to rare events, using data on Prussian cavalry deaths from horse kicks
Kingman, J. F. C. (1993), Poisson Processes, Oxford Studies in Probability 3, Oxford: Clarendon Press — the standard reference on Poisson processes (independent increments, exponential inter-arrival times)
Cameron, A. C. & Trivedi, P. K. (2013), Regression Analysis of Count Data (2nd ed.), Cambridge: Cambridge University Press — the standard textbook on count data, covering the limits of the Poisson model (overdispersion, zero inflation) and its alternatives
