Poisson Distribution

Historical Context and Assumptions

The Poisson distribution is attributed to the prolific 19th – century French mathematician Siméon Denis Poisson, although it was known by the probabilist Abraham de Moivre well over a century earlier. Poisson’s discovery was chiefly motivated as an approximation to the binomial distribution, described in the next section.

Technical details aside, the number of occurrences of an event over a given period of time can be said to follow a Poisson distribution with rate λ if four basic assumptions are met. Suppose an experimenter is going to observe some phenomenon over time (or other continuous unit). The basic Poisson assumptions are as follows:

• 1.
  The probability of observing one event in a short period of time is approximately equal to the rate λ times the duration of the period.
• 2.
  The likelihood of observing two or more events in a short interval is approximately zero.
• 3.
  The probability of observing j events in one time period and k events in a separate time period is equal to the product of those probabilities individually.
• 4.
  The rate referred to in (1) does not change over time.

These assumptions have been used in the past to justify many applications, including (famously) [Page 1256]the number of soldiers killed per year by horse kicks, eye movements of various types per minute while reading, and the defects in manufactured magnetic tape per yard. These ideas are generalized by the notion of a Poisson process, an important stochastic process studied in probability and statistics.

Mathematical Properties

A random variable Y with the Poisson (λ) distribution, denoted Y ~ Pois(λ), assigns a potential outcome y = 0, 1, 2, … , probability following the Poisson formula:

$$f(y)=P[Y=y]=\frac{e^{-\lambda }{\lambda }^y}{y!},$$

where λ > 0 is the rate constant, e is Euler’s exponential constant (roughly equal to 2.72), and y! = y(y−1)(y–2) … (2)(1) is the factorial of the positive integer y and 0!, defined to be 1. Its mean (expected value) and variance are both equal to the rate parameter λ, and consequently, the larger the expected number of counts, the larger those counts are expected to vary across repeated sampling. The distribution is characteristically right skewed, as shown in Figure 1; however, the skewness diminishes as λ grows.

Figure 1 Probabilities assigned by Poisson distributions with λ = 2, 5, and 10

The Poisson distribution is closely related to the binomial distribution through a mechanism referred to as the law of small numbers or rare events. Specifically, the probabilities assigned to the numbers 0, 1, 2, … , n by a binomial distribution with parameters n and p, ${}_n{C}_y{p}^y{\left(1-p\right)}^{n-y}$, converge to those assigned by the Poisson distribution with λ = np as n→∞ and p→0, with n and p “balancing out” to the fixed positive constant λ. As a consequence, the Poisson probability formula can be used to approximate that of the binomial; it is typically recommended that n be at least 20 and p be less than 5%.

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