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Poisson and exponential distribution

The Poisson distribution models the number of events in a fixed time interval, such as the number of failures per hour. The exponential distribution models the time between such events, and is often used for lifetime with a constant failure rate. The two are linked: if events are Poisson distributed, the time between them is exponentially distributed.

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Poisson: probability of

k

events

P(T > t) = e^{-t/\theta}

exponential: probability of lifetime beyond

t

Symbols

$\lambda$	average number of events per interval
$\theta$	expected lifetime

Example

$\lambda = 4$ failures per hour: $P(X=2) = \dfrac{4^2 e^{-4}}{2!} \approx 0.1465$ .

Expected lifetime 1000 h: $P(T > 500) = e^{-0.5} \approx 0.6065$ .

Poisson counts events, exponential measures the time between them — both are governed by the same rate.

Practise distributions for free →

← Binomial distribution · Mean, median and standard deviation →

Part of Statistics and Risk Analysis: Distributions.