IB Maths AI 4.12 Notes

Poisson distribution

The Poisson distribution is used to model the number of times an event happens in a fixed interval of time, length, area, or volume. It is appropriate when:

• events occur independently
• the events happen at a constant average rate over the interval being considered

Typical examples include the number of calls arriving at a help desk in one hour, the number of typing errors on a page, or the number of cars passing a point in one minute.

If $X$ follows a Poisson distribution with parameter $\lambda$, we write:

$X \sim \text{Po}(\lambda)$

where $\lambda$ is the expected number of events in the interval.

The probability formula is:

$P(X=r)=\frac{e^{-\lambda}\lambda^r}{r!}$

for $r=0,1,2,\dots$

Here, $\lambda$ is the mean number of events, $e$ is the constant base of natural logarithms, and $r$ is the number of events required.

For a Poisson random variable, $E(X)=\lambda$ and $\mathrm{Var}(X)=\lambda$. So for a Poisson distribution, the mean and the variance are equal.

Suppose the number of emails received in an hour follows a Poisson distribution with mean $4$. Find the probability of receiving exactly $2$ emails.

$X\sim\text{Po}(4)$.

$P(X=2)=\frac{e^{-4}4^2}{2!}=\frac{16e^{-4}}{2}$

$P(X=2)=8e^{-4}\approx0.147$.

The probability of receiving exactly $2$ emails is $\approx0.15$.