Poisson Distribution
The Poisson distribution models the number of events occurring within a fixed interval of time or space, given that events happen independently at a constant average rate λ. It is a discrete distribution and is particularly useful when events are rare relative to the number of opportunities. Applications include modelling call-centre traffic, radioactive decay counts, and the number of defects per unit area.
Key Formula
P(X = k) = (λ^k × e^(−λ)) / k!
LaTeX: P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
| Symbol | Meaning | Unit |
|---|---|---|
| λ | average number of events per interval (rate parameter) | events per interval |
| k | actual number of events (k = 0, 1, 2, ...) | unitless |
| e | Euler's number ≈ 2.71828 | unitless |
| k! | factorial of k | unitless |
Worked Example
Problem
A call centre receives on average 4 calls per minute. What is the probability of receiving exactly 6 calls in a given minute?
Solution
Step 1: Identify λ = 4, k = 6. Step 2: Apply the formula: P(X = 6) = (4^6 × e^−4) / 6! Step 3: Calculate 4^6 = 4096. Step 4: Calculate e^−4 ≈ 0.01832. Step 5: Calculate 6! = 720. Step 6: P(X = 6) = (4096 × 0.01832) / 720 ≈ 75.1 / 720 ≈ 0.1042.
Answer
P(X = 6) ≈ 0.1042, or about 10.4%
Properties of the Poisson Distribution
| Property | Formula | Description |
|---|---|---|
| Mean | μ = λ | Average number of events per interval |
| Variance | σ² = λ | Equal to the mean (unique property) |
| Std Deviation | σ = √λ | Square root of the rate |
| Skewness | 1/√λ | Positive skew, decreases as λ increases |
| Mode | floor(λ) or floor(λ)−1 | Most probable count |
| Approx. Normal | λ > ~10 | Approaches normal for large λ |
Interactive Tools
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Related Terms
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is a discrete probability distribution characterised by two parameters: n (number of trials) and p (probability of success on each trial). It is widely used in quality control, clinical trials, and survey analysis.
Probability Distribution
A probability distribution is a mathematical function that describes the likelihood of each possible outcome of a random variable. It assigns a probability to every possible value or range of values that the variable can take, with all probabilities summing to 1. Probability distributions are foundational in statistics and are used in fields ranging from insurance and finance to physics and machine learning.
Expected Value
The expected value (or expectation) of a random variable is the probability-weighted average of all possible values it can take. It represents the long-run average outcome if the experiment were repeated many times. Expected value is central to decision theory, gambling, insurance, and financial risk analysis.
Named after French mathematician Siméon Denis Poisson, who introduced it in his 1837 work Recherches sur la probabilité des jugements. The word Poisson means "fish" in French, though the distribution is named after the person, not the animal.