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 [latex](RV)[/latex] 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 [latex]\mu[/latex] of the event in the interval. Notation: [latex]X \sim P(μ)[/latex].

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