Multiple choiceQuestion 1 · Recall
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By the end you can state and use the facts that a Poisson distribution Po(λ) has mean λ and variance λ, decide whether a Poisson model is plausible for a real situation, and scale a rate from one interval to another.
If X ~ Po(λ) then
E(X) = λ and Var(X) = λ
so the standard deviation is √λ. (Here E(X) is the mean, or expected value, Var(X) is the variance, and the standard deviation is the square root of the variance.) This is unusual: most distributions need one parameter to fix the mean and another to fix the spread, but here a single number λ does both. λ is the average number of events in the interval you are counting in.
Two consequences are worth memorising because examiners use them constantly:
For example, for λ = 4 the variance is 4 and the standard deviation is √4 = 2, half the mean.
The Poisson model
A random variable X has a Poisson distribution with parameter λ, written X ~ Po(λ), when X counts the number of events in a fixed interval and
λ is the mean number of occurrences in that interval. The interval may be one of time, length, area or volume, and it must be fixed in advance: λ for 2 minutes is not the same as λ for 1 minute.
Worked example: Example 1: nothing to calculate
The number of emails arriving at an office in a 10-minute period is modelled as X ~ Po(3.5). State the mean, the variance and the standard deviation of X.