Poisson distribution
The Poisson distribution counts events in a fixed time or space when they happen at a steady average rate lambda and independently. P(X = k) = e^(-lambda) x lambda^k / k!. Mean and variance both equal lambda.
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Analogy
Emails arriving in your inbox: you know the hourly average, but you cannot say when each one comes.
Example
Average lambda = 4 per hourP(X=2) = e^-4 x 4^2 / 2= 0.01832 x 8 = 0.1465P(X=0) = e^-4 = 0.0183
In practice
Models calls to a help desk, typos per page and cars passing a point per minute.
Practice
Flashcards
- Mean and variance of a Poisson?Both equal lambda.
Questions people ask
When should I use Poisson instead of binomial?
Use Poisson when you count events over time or space and there is no fixed number of trials, only an average rate. Use binomial when there are n trials with a set success chance. For huge n and tiny p, the binomial is well approximated by a Poisson.
What happens to lambda if I change the time window?
Lambda scales with the window. If calls average 4 per hour, then lambda is 2 for a 30-minute window and 8 for two hours. Always match lambda to the interval in the question before using the formula, because a wrong window gives a confidently wrong answer.