Poisson Distribution

What Is the Poisson Distribution?

The Poisson distribution is a probability distribution that describes how many times an event is likely to occur within a fixed period or area, assuming the events happen at a steady average rate and each one occurs independently of the others. It is named after the French mathematician Siméon-Denis Poisson. Its single parameter, usually written as lambda (λ), is the average number of events expected in that interval.

Because it models counts of discrete occurrences, the distribution appears in many fields. In finance and economics, analysts use it to model things such as the number of defaults in a loan pool, the arrival of large trades, or sudden market jumps over a given window. Defaults are one form of credit risk, which is why the model is often applied there.

How It Works

The probability of observing exactly k events is:

P(k) = (λ^k × e^(−λ)) / k!

Here e is the mathematical constant of about 2.718, and k! is k factorial. The mean of a Poisson distribution, which is its expected value, equals λ. The variance also equals λ. That equality is a useful check when deciding whether the model fits a set of data.

An Example

Suppose a trading desk historically logs an average of 2 erroneous trade reports per day, so λ = 2. Under this model, the chance of zero errors on a given day is about 13.5%, and the chance of exactly three errors is about 18.0%. Probabilities for high counts fall off quickly, which is why the distribution suits rare events.

Limitations for Investors

The model rests on two assumptions that often fail in real markets: a constant average rate and independence between events. Market stress tends to cluster, so events may arrive in bursts, and the actual spread of outcomes can be wider than a Poisson model predicts. For this reason, analysts often compare it with alternatives such as the negative binomial distribution, which allows greater variability. Periods of high volatility are a common case where the simple model understates risk. The Poisson distribution is a modeling tool, not a forecast of when any particular event will happen.