The Poisson distribution

6.1 The Poisson distribution

When to use it

$X \sim \mathrm{Po}(\lambda)$ counts how many events happen in a fixed interval of time or space, such as calls per hour or flaws per metre of cloth.

The model fits when events occur:

• at random
• independently of each other
• singly, not at exactly the same moment
• at a constant average rate

Probabilities

$\mathrm{P}(X = 2) = \mathrm{e}^{-3}\frac{3^2}{2} = 0.224$

$\mathrm{P}(X \leq 1) = \mathrm{e}^{-3}(1 + 3) = 0.199 \qquad \mathrm{P}(X \geq 2) = 1 - 0.199 = 0.801$

Mean and variance

So a Poisson model is plausible only when the mean and variance of the data are close. A big difference between them suggests that it does not fit.

They are close, so this supports using $\mathrm{Po}(3.1)$.

Scaling the interval

The mean grows in proportion to the size of the interval.

For 30 minutes, $\lambda = 1$, so $\mathrm{P}(0) = \mathrm{e}^{-1} = 0.368$.

For 3 hours, $\lambda = 6$, so $\mathrm{P}(5) = \mathrm{e}^{-6}\dfrac{6^5}{5!} = 0.161$.

Using the Poisson for a binomial

For a large number of trials with a small probability, the two distributions are close.

The working is much shorter this way, since the binomial needs very large and very small numbers.

$X \sim \mathrm{B}(200, 0.01)$, with $n > 50$ and $np = 2 < 5$, so use $\mathrm{Po}(2)$.

$\mathrm{P}(X \leq 1) = \mathrm{e}^{-2}(1 + 2) = 0.406$

Using the normal for a Poisson

For a large mean, the Poisson distribution becomes bell-shaped.

Use $Y \sim \mathrm{N}(20, 20)$, and $X \geq 25$ becomes $Y > 24.5$:

$z = \frac{24.5 - 20}{\sqrt{20}} = 1.006 \qquad 1 - \Phi(1.006) = 0.157$

Choosing an approximation

Finding a value

Some questions give a probability and ask for $\lambda$.

$\mathrm{e}^{-\lambda} = 0.2$, so $\lambda = -\ln 0.2 = 1.61$.