The Poisson distribution

Need to know

What you need to know

• Events must occur **singly** in time or space. This means two events cannot happen at the exact same instant.
• Events must occur **independently**. The occurrence of one event does not affect the probability of another event occurring.
• Events must occur at a **constant average rate**. The mean number of events in an interval is proportional to the size of the interval. For example, if the average is 2 events per hour, it's 1 event per 30 minutes.

Explanation

Counting Rare Events

1. Poisson models events in a fixed interval, like time or space, where the average number of events, λ, is known. The events must be rare and independent.
2. The probability of observing exactly 'r' events is given by the formula P(X = r) = e^{−λ} λ^r / r!. You'll typically use your calculator's function or statistical tables for this.
3. A unique and crucial property is that the mean (expected value) and the variance are both equal to λ. So, E(X) = Var(X) = λ.
4. When n is large and p is small, the Poisson distribution can approximate the binomial distribution. We set the Poisson mean λ equal to the binomial mean, np.