The Poisson distribution as a model for random events, its mean and variance, the binomial distribution, the additive property of Poisson variables, and the Poisson approximation to the binomial.

What this dot point is asking

Edexcel Further Statistics wants you to use the Poisson distribution to model random events over a fixed interval, state and apply its mean and variance, use the binomial distribution, add independent Poisson variables, and apply the Poisson approximation to the binomial when $n$ is large and $p$ small. Justifying the choice of model and approximation, with clear method, is what earns the marks.

The Poisson distribution

The Poisson models the number of random events occurring in a fixed interval of time or space, when events happen independently at a constant average rate $\lambda$. The distinctive feature is that its mean and variance are equal, both $\lambda$, which is also the practical test of whether a Poisson model is plausible for a data set: if the sample mean and variance are close, a Poisson fit is reasonable.

The binomial and the additive property

The binomial distribution counts the number of successes in a fixed number $n$ of independent trials, each with the same success probability $p$. Independent Poisson variables have a clean additive property: their sum is again Poisson, with the rates added, which lets you combine, for example, the number of events over two separate intervals into a single Poisson model.

The Poisson approximation to the binomial

When the number of trials is large and the success probability is small, the binomial probabilities are awkward to compute but the Poisson distribution with $\lambda = np$ gives a very close approximation. The rule of thumb is $n$ large (commonly $n > 50$) and $p$ small (commonly $p < 0.1$), so that $np$ is moderate.

Examples in context

The Poisson and binomial are the two most heavily examined discrete models. Their means and variances are derived using the expectation machinery of the discrete-distributions dot point. They contrast with the geometric and negative binomial "waiting time" models, which count trials rather than successes. Both distributions are the usual subjects of chi-squared goodness of fit tests, where the mean is often estimated from the data (costing a degree of freedom). The additive property of the Poisson reflects the way independent rates combine, and the approximation to the binomial is the historical origin of the Poisson distribution as the "law of rare events".

Try this

Q1. For $X \sim \operatorname{Po}(3)$, find $P(X = 0)$. [2 marks]

• Cue. $P(X = 0) = e^{-3} \approx 0.0498$.

Q2. State the mean and variance of $B(50, 0.2)$. [2 marks]

• Cue. Mean $= 10$, variance $= 50(0.2)(0.8) = 8$.

Q3. Independent $X \sim \operatorname{Po}(2)$ and $Y \sim \operatorname{Po}(6)$. State the distribution of $X + Y$. [1 mark]

• Cue. $X + Y \sim \operatorname{Po}(8)$.