Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of times a random event occurs within a fixed interval of time, space, or any other measurable quantity. It is named after the French mathematician Siméon Denis Poisson, who introduced it in 1837. Unlike the binomial distribution, which counts successes in a fixed number of trials, the Poisson distribution is most useful when events happen at a constant average rate and independently of one another — for example, the number of customers arriving at a bank per hour, the number of radioactive decays per second, or the number of misprints on a page. The distribution is characterized by a single parameter, λ (lambda), which simultaneously represents both the mean and the variance of the distribution. This elegant property makes the Poisson distribution mathematically convenient and easy to apply. When deciding which distribution to use in a problem, recognizing Poisson conditions is essential: events must occur independently, at a constant average rate, and the probability of two events occurring at exactly the same instant must be negligible. If these conditions hold, the Poisson model is appropriate regardless of whether the underlying process involves time, area, volume, or length. One of the most powerful results in probability theory concerning the Poisson distribution is the additive property: if X and Y are independent Poisson random variables with parameters λ₁ and λ₂ respectively, then their sum X + Y also follows a Poisson distribution with parameter λ₁ + λ₂. This makes the Poisson distribution especially useful in combining independent event streams, such as calls arriving from two different regions of a city. Understanding the Poisson distribution, its properties, and the conditions under which it applies prepares students for a wide range of real-world statistical modelling problems.

Key takeaways
  • The Poisson Probability Formula
  • Mean and Variance
  • Conditions for Using the Poisson Distribution
  • Additive Property of Independent Poissons
  • Selecting Between Distributions

The Poisson Probability Formula

If X ~ Po(λ), then the probability that X takes a specific non-negative integer value r is given by P(X = r) = (e^{-λ} · λ^r) / r!, where r = 0, 1, 2, 3, … The parameter λ > 0 is the mean number of occurrences in the given interval. The factor e^{-λ} ensures that all probabilities sum to 1 over all non-negative integers, which can be verified using the Taylor series expansion of e^λ.

Mean and Variance

A defining and elegant feature of the Poisson distribution is that both its mean (expected value) and variance are equal to λ. That is, E(X) = λ and Var(X) = λ. This means the standard deviation is √λ. In practice, if you observe a data set and notice that the sample mean and sample variance are approximately equal, this is a strong indicator that a Poisson model may be appropriate. For example, if λ = 4, then on average 4 events occur per interval, and the standard deviation of the count is √4 = 2.

Conditions for Using the Poisson Distribution

Four key conditions must be satisfied for the Poisson model to be appropriate: (1) Events occur singly — the probability of two or more events in a very short interval is negligible. (2) Events occur independently — the occurrence of one event does not affect the probability of another. (3) The average rate λ is constant over the interval being considered. (4) The interval is fixed — it could be a fixed time period, area, volume, or length. When these conditions hold, the Poisson distribution is the correct model.

Additive Property of Independent Poissons

If X ~ Po(λ₁) and Y ~ Po(λ₂) are independent random variables, then X + Y ~ Po(λ₁ + λ₂). This result extends to any finite number of independent Poisson variables: X₁ + X₂ + … + Xₙ ~ Po(λ₁ + λ₂ + … + λₙ). This property is extremely useful in practice. For instance, if accidents at junction A occur at a rate of 2 per month and at junction B at a rate of 3 per month, and these are independent, the total number of accidents at both junctions follows Po(5).

Selecting Between Distributions

When choosing between the Binomial distribution B(n, p) and the Poisson distribution Po(λ), ask: Is the number of trials n fixed and finite, and is each trial a clear success/failure? If yes, use Binomial. Are events occurring at a constant rate with no fixed upper bound on the count? If yes, use Poisson. The Poisson distribution can also be used as an approximation to the Binomial when n is large (n ≥ 50) and p is small (p ≤ 0.1), using λ = np. This approximation works well because rare, independent events behave like a Poisson process.

The Key Equation

P(X=r)=e−λ λrr!,r=0,1,2,…P(X = r) = \frac{e^{-\lambda}\,\lambda^{r}}{r!}, \quad r = 0, 1, 2, \ldots
  • P(X = r) — Probability (dimensionless): The probability that the random variable X takes the exact value r.
  • e — Euler's number (dimensionless): The mathematical constant approximately equal to 2.71828, the base of the natural logarithm.
  • \lambda — Rate parameter (lambda) (events per interval): The mean number of events in the given interval; also equal to the variance of the distribution.
  • r — Number of occurrences (dimensionless (count)): The specific non-negative integer outcome whose probability is being calculated.
  • r! — r factorial (dimensionless): The product r × (r−1) × … × 2 × 1, representing the number of ways to arrange r items; by convention 0! = 1.

Worked Example

Problem. A hospital emergency department receives, on average, 4 patients per hour. Assuming arrivals follow a Poisson distribution, find: (a) P(X = 2), (b) P(X ≤ 2), (c) the probability that in a 2-hour period fewer than 3 patients arrive.

P(X=2)≈0.1465,P(X≤2)≈0.2381,P(Y<3)≈0.0137P(X=2)\approx0.1465,\quad P(X\leq2)\approx0.2381,\quad P(Y<3)\approx0.0137

Answer. P(X=2) ≈ 0.1465; P(X ≤ 2) ≈ 0.2381; P(fewer than 3 in 2 hours) ≈ 0.0137

Figure
Diagram
Poisson PMF: Probability P(X = r) for λ = 2 and λ = 5r (number of events)P(X = r)0123456789101100.10.20.30.4λ = 2λ = 5
Origins · 1 minSiméon Denis Poisson and how this was discovered
SD
Siméon Denis Poisson
1781–1840 · French

Poisson introduced the Poisson distribution in his 1837 work 'Recherches sur la probabilité des jugements en matière criminelle et en matière civile' (Research on the Probability of Judgments in Criminal and Civil Matters). He derived it as a limiting case of the binomial distribution when n is large and p is small, a result now known as the law of rare events. His work laid foundations for modern probability theory and mathematical physics. Poisson is said to have declared that 'Life is good for only two things: doing mathematics and teaching it.' He also made major contributions to mathematical physics, including Poisson's equation in electrostatics and gravitational theory, which today appear in every university physics course.

  • Binomial Distribution — The Poisson distribution can be derived as a limiting case of the binomial distribution when n is large and p is small, making comparison between these two models essential for selecting the correct distribution in any problem.
  • Expectation and Variance — The mean E(X) = λ and variance Var(X) = λ of the Poisson distribution are derived using the general formulas for expectation and variance of discrete random variables, which are prerequisite concepts.
  • Normal Approximation to Distributions — For large λ, the Poisson distribution is well approximated by a Normal distribution with mean λ and variance λ, providing a powerful computational tool and connecting discrete and continuous probability models.
  • Discrete Random Variables — The Poisson distribution is a discrete random variable whose probability mass function, cumulative probabilities, mean, and variance are all computed using the general framework of discrete random variable theory.