Poisson Distribution

Definition

The Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed interval of time or space when events occur independently and at a constant average rate. Its single parameter, λ (lambda), represents the expected number of events in that interval.

Formula

The probability of exactly x events is
f(x) = λ^x e^(−λ) / x!
where
* x = number of occurrences (0, 1, 2, …)
* λ = mean number of occurrences in the interval
* e = Euler’s number (~2.71828)
* x! = factorial of x

Example: if λ = 3, the probability of zero events is f(0) = e^(−3) ≈ 0.0498.

Key properties

• Discrete: counts only whole numbers (0, 1, 2, …).
• Mean = Variance = λ.
• Events must be independent, occur at a constant average rate, and not occur simultaneously.
• For large λ, the Poisson distribution is approximately normal (by the central limit idea).

When to use

Use the Poisson distribution when:
* The variable is a count (how many times something happens).
* Events are rare or occur sporadically over a defined interval.
* You can reasonably assume independence and a constant event rate.

Common applications

• Operations and service planning (calls, arrivals, defects) — helps size staffing or inventory for expected counts.
• Reliability and safety (failures, accidents) — model counts of incidents in a period.
• Biology and medicine (cell counts, mutation events).
• Finance — modeling counts such as number of trades per period, number of defaults or shocks to a market over a horizon, or rare-event jump counts in asset models.

Practical notes

• To get probabilities of “at least” or “more than” a number of events, sum the relevant f(x) values or use cumulative Poisson functions available in statistical software.
• If observed variance substantially exceeds the mean, data may be overdispersed and alternatives (e.g., negative binomial) may be more appropriate.

Explain like I’m 5

Imagine counting how many times a rare bird visits your yard each week. If on average three birds come each week and visits are independent, the Poisson distribution tells you how likely you are to see 0, 1, 2, 3, or more birds in a given week.

Bottom line

The Poisson distribution is a simple, single-parameter model for count data occurring randomly over a fixed interval. Its mean-equals-variance property and discrete nature make it a natural choice for modeling rare or countable events in operations, science, and finance when the underlying assumptions hold.