Poisson Distribution
Maths: Statistics for machine learning
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Published Oct 22 2025, updated Aug 17 2026
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Machine LearningMathsNumPyPandasPythonStatistics
The Poisson distribution models the number of times an event occurs in a fixed interval of time, area, or volume,
given that:
- The events occur independently,
- The average rate (λ) is constant, and
- Two events cannot happen at exactly the same instant.
In short:
It tells you the probability of getting k events in a given interval when you know the average rate λ (lambda).
Formula: Probability Mass Function (PMF)
Where:
- X = number of events
- λ = average number of occurrences (mean rate)
- e = 2.718 (Euler’s number)
The total probability adds up to 1:
Examples:
- Customer arrivals - Number of customers entering a store per hour
- Emails received - Emails per minute
- Defects in manufacturing - Defective parts per batch
- Website hits - Visits per second
- A bar chart where the probability peaks around λ (mean)
- The distribution is right-skewed for small λ values (rare events)
- As λ increases, it becomes more symmetrical and approaches the Normal distribution
Example (λ = 4):
- Most likely outcomes: k = 3, 4, or 5
- Low probability for 0 or large counts
In Machine Learning
- Count data modelling - Number of clicks, visits, failures, or events per interval
- Anomaly detection - Unusually high event counts signal rare anomalies
- Natural language processing (NLP) - Word occurrence counts in documents
- Regression - Poisson Regression models count-based target variables
- Queueing theory / Simulation - Modelling arrivals, traffic, or process events