Used to model counts
\[ P(X = x;\lambda) = \frac{\lambda^x e^{-\lambda}}{x!}\]
Where \(x\) is defined in the non-negative integers.
The mean of this distribution is \(\lambda\).
The variance of this distribution is also \(\lambda\)
Contingency tables are used when you classify for example people following their hair color by race. An example of contingency table would be
| Body | red | yellow | magenta |
|---|---|---|---|
| black | 1 | 0 | 0 |
| blue | 0 | 1 | 0 |
| green | 0 | 0 | 1 |
So the Poisson distribution is the default distribution for modeling contingency table data. And it turns out that it has a deep connection with some of the other models such as multinomial, binomials, and Bernoullis
The last use of the Poisson is done quite often in epidemiology. Where \(n\), the population, is so large, but \(p\), or the events, are not so common.
Poisson variables are used to model rates
\(X \sim Poisson(\lambda t)\) where
\(\lambda = E[X/t]\) is the expected count per unit of time
\(t\) it the total monitoring time
The number of people that show up at bus stop is Poisson with a mean of \(2.5\) per hour. If watching the bus stop for \(4\) hours, what is the probability that \(3\) or fewer people show up for the whole time?
ppois(3, lambda = 2.5 * 4)## [1] 0.01033605When \(n\) is large and \(p\) is small the Poisson distribution is an accurate approximation to the binomial distribution
Notation
\(X \sim Binomial(n, p)\)
\(\lambda = np\)
\(n\) gets large
\(p\) gets small
We flip a coin with success probability \(0.01\) five hundred times. What’s the probability of 2 or fewer successes?
The binomial version
pbinom(2, size = 500, prob = 0.01)## [1] 0.1233858The poisson approximation with \(\lambda = np\)
ppois(2, lambda = 500 * 0.01)## [1] 0.124652