The Poisson distributionEdexcel International A Level Further Maths: Revision notes

A Poisson distribution models the number of times $X$ an event occurs in a fixed interval of time, length or area, when events happen randomly at a constant average rate. We write $X\sim\text{Po}(\lambda)$, where $\lambda$ is the mean number of events in the interval. The probability function is $P(X=r)=e^{-\lambda}\frac{\lambda^r}{r!},\quad r=0,1,2,\ldots$ Three conditions are needed: events occur independently of each other, they occur singly (never two at the same instant), and the average rate is constant. Typical examples are flaws in a roll of cloth, calls to a switchboard, or misprints per page.

For a single value use the formula or the calculator. For cumulative probabilities, use the Poisson tables in the formulae booklet or add the individual terms. The tables give $P(X\le r)$, so rewrite other inequalities:

• $P(X<r)=P(X\le r-1)$
• $P(X\ge r)=1-P(X\le r-1)$
• $P(X>r)=1-P(X\le r)$
• $P(a\le X\le b)=P(X\le b)-P(X\le a-1)$ Example: $X\sim\text{Po}(4)$. $P(X=3)=e^{-4}\frac{4^3}{3!}=0.1954$. $P(X\le3)=0.4335$, so $P(X\ge4)=1-0.4335=0.5665$.

For $X\sim\text{Po}(\lambda)$ the mean and the variance both equal the parameter: $E(X)=\lambda,\qquad\text{Var}(X)=\lambda.$ No derivation is needed. The standard deviation is $\sqrt\lambda$. If a data set has a sample mean and variance that are about equal, that supports a Poisson model; a variance much larger than the mean suggests the events are clustered, so the model is poor.

If events occur at rate $\lambda$ per unit, then in $k$ units the number of events is $\text{Po}(k\lambda)$. For example, 3 calls per minute becomes $\text{Po}(15)$ for 5 minutes. The additive property: if $X\sim\text{Po}(\lambda)$ and $Y\sim\text{Po}(\mu)$ are independent, then $X+Y\sim\text{Po}(\lambda+\mu)$. This is useful for combining areas or periods, such as flaws in two separate pieces of fabric.

Exam questions often ask whether a Poisson model is appropriate. Compare each condition with the context:

• Independence: could one event cause another, for example a breakdown causing a queue?
• Singly: can events arrive in batches, such as a coach party arriving together?
• Constant rate: does the rate change with the time of day, so that rush hour differs from midday? A good answer names the condition and says how the context breaks it, for example 'buses follow a timetable so arrivals are not random'. State the assumption when you use the model: 'assuming flaws occur at random at a constant rate'.