The Poisson distributionEdexcel A-Level Further Maths: Revision notes

The Poisson distribution models the number of events occurring in a fixed interval of time or space. If $X\sim\text{Po}(\lambda)$, where $\lambda>0$ is the mean number of events in the interval, then $P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!},\quad x=0,1,2,\ldots$ There is no upper limit on $X$. The model is suitable when events occur singly, independently of each other and at a constant mean rate. Examples are emails per 10 minutes, faults per metre of cloth and calls per hour. Unlike a binomial distribution, there is no fixed number of trials.

Use the Poisson function on your calculator for $P(X=x)$ and the cumulative function for $P(X\leq x)$. Convert other inequalities to these forms. For $X\sim\text{Po}(3.5)$:

• $P(X=2)=0.185$
• $P(X\leq2)=0.3208$
• $P(X\geq3)=1-P(X\leq2)=0.679$
• $P(X>3)=1-P(X\leq3)$, which equals $P(X\geq4)$
• $P(2<X\leq6)=P(X\leq6)-P(X\leq2)=0.614$ The key is to get the boundary right: $P(X\geq k)=1-P(X\leq k-1)$.

The mean rate scales with the length of the interval. If events occur at a mean rate of $\lambda$ per minute, the number in $t$ minutes is $\text{Po}(t\lambda)$. For example, 6 calls per hour gives $\text{Po}(2)$ for 20 minutes and $\text{Po}(1)$ for 10 minutes. Always match the time unit in the question before using the formula. Then separate periods are independent, so the probability of events in two periods is the product: the probability of exactly 1 call in each of two 20-minute periods is $(2e^{-2})^2=0.0733$. When counting several periods, a binomial distribution may be used for how many periods satisfy a condition.

If $X\sim\text{Po}(\lambda)$ and $Y\sim\text{Po}(\mu)$ are independent, then $X+Y\sim\text{Po}(\lambda+\mu)$. No proof is required. For example, if online orders are $\text{Po}(3)$ and phone orders are $\text{Po}(2)$ per hour, independently, the total is $\text{Po}(5)$ per hour. Use it to combine sources of events, then rescale for the interval needed. The property needs independence: if the two sources are linked, the total is not Poisson in general.

To model a situation, choose $\lambda$ as the observed mean rate for the interval. To comment critically, check each condition in context. Independence: does one event make another more or less likely (convoys, queues, epidemics)? Constant rate: does the rate change within the period (rush hour, evening peaks)? Singly: can events happen in groups (a coach arriving with several passengers)? Name the condition that fails and explain in terms of the context. Also say what the probability means in context, for example 'there is a 9.4% chance that more than 5 vehicles pass in a minute'.