The Poisson Distribution

A discrete distribution for non-negative integers in statistics and probability

The Poisson distribution is a discrete probability distribution whose random variable represents the number of events that occur in a given time interval. It makes several key assumptions:

• The number of events can be represented by non-negative integers (0, 1, 2, 3, …).

• The events occur at a constant rate throughout any interval of time.

• Two events cannot occur at the same instant of time.

• The occurrence of one event does not affect the probability of the occurrence of another event. In other words, any two events are probabilistically independent.

So far, I have described the events occurring over time. You can use some other unit for denominator, like area or volume.

The probability mass function (PMF) is

\(P(X=x) = \frac{\lambda^x e^{-\lambda}}{x!}, \ \ \ x = 0, 1, 2, 3, ...\)

Note that the support set is all non-negative integers. It is important to note that there is no upper limit to the support set; the value of X can be infinitely large.

The expected value of X is simply the rate parameter, λ.

\(E(X) = \lambda\)

The variance of X is also the rate parameter, λ. This makes the Poisson distribution special: The expected value and variance are the same for X.

\(V(X) = \lambda\)

Using some algebra, you can reach both equations by applying the definitions of expectation and variance to X.

---

The Poisson distribution is common in many business and scientific applications, such as:

• The number of shoppers that go into a store

• The number of cars that pass through an intersection

• The number of photons that reach a telescope