Key Concepts

• A Poisson distribution is a discrete random variable.
• Calculating a Poisson probability is based on the average number of occurrences for a specific interval of time.
• For a Poisson distribution, the chances of the event happening are independent of when the event previously happened.

Glossary

Poisson probability distribution: a discrete random variable $(RV)$ that counts the number of times a certain event will occur in a specific interval; characteristics of the variable:

• The probability that the event occurs in a given interval is the same for all intervals.
• The events occur with a known mean and independently of the time since the last event.

The distribution is defined by the mean $\mu$ of the event in the interval. Notation: $X \sim P(μ)$.

This mean is $\mu =np$. The. standard deviation is $\sigma = \sqrt{\mu}$. The probability of having exactly $x$ successes. in $r$ trials is $P(X=x)=e^{- \mu} (\frac{\mu^{x}}{x!})$. The Poisson distribution is often used to approximate the binomial distribution, when $n$ is “large” and $p$ is “small” (a general rule is that $n$ should be greater than or equal to 20 and $p$ should be less than or equal to 0.05).