4.5 Poisson distribution

Poisson distribution is often useful for estimating the number events in a large population over a unit of time.

The *rate* for a Poisson distribution is the average number of occurrences in a mostly-fixed population per unit of time. The parameter is the rate (i.e. how many events we expect to observe) and it is typically denoted by \(\lambda\) or \(\mu\). Using the rate, we can describe the probability of observing \(k\) events in a single unit of time.

[!def] Poisson Distribution Suppose we are watching for events and the number of observed events follows a Poisson distribution with rate \(\lambda\). Then \(\large P(\text{observe }k \text{ events})=\frac{\lambda^k e^{-\lambda}}{k!}\) The mean and standard deviation of this distribution are \(\lambda\) and \(\sqrt{\lambda}\)

A random variable may follow a Poisson distribution if we are looking for the number of events, the population that generates such events is large, and the events occur independently of each other.

Some events that are not really independent can still follow a Poisson model - i.e. weekends are more popular for weddings. The idea of modeling rates for a Poisson distribution against a second variable such as the day of the week forms the foundation of some methods of generalized linear models.