Revise The Poisson distribution for Cambridge 9709 (6.1)

For a random variable to be modelled by a Poisson distribution, certain conditions must be met. These are crucial for justifying its use in an exam context. The events must be occurring in a fixed, continuous interval of time or space.

If a random variable $X$ follows a Poisson distribution with a mean of $\lambda$, we write this as $X \sim Po(\lambda)$. The probability of observing exactly $r$ events in the interval is given by a specific formula. While the formula is important, in practice you will often use the statistical functions on your calculator or tables provided in the exam.

If $X \sim Po(\lambda)$, then the probability of $r$ occurrences is: $P(X=r) = \frac{e^{-\lambda} \lambda^r}{r!}$ \quad for $r = 0, 1, 2, ...$

Your calculator's distribution mode will have a 'Poisson PD' (Probability Density) for calculating $P(X=r)$ and a 'Poisson CD' (Cumulative Distribution) for calculating $P(X \le r)$. Mastering the cumulative function is key to solving questions involving inequalities like 'at least' or 'more than'.

A remarkable and unique feature of the Poisson distribution is that its mean and variance are the same, both equal to the parameter $\lambda$. This property is often tested and can be used to justify whether a Poisson model is appropriate for a given set of data. If the sample mean and sample variance are not close, the model may be unsuitable.

If $X \sim Po(\lambda)$, then: \nMean: $E(X) = \lambda$ \nVariance: $Var(X) = \lambda$

This distribution also serves as a useful approximation for the Binomial distribution, $X \sim B(n, p)$, when certain conditions are met. This is particularly helpful when $n$ is very large, as calculating binomial probabilities directly can be computationally intensive.