Waiting Time for Poisson Process obeys Exponential Distribution

Theorem

Let $P$ be a Poisson process in which events occur at a rate of $k$ per unit time.

Then the waiting times of $P$ are exponentially distributed with expectation $\dfrac 1 k$.

Proof

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Examples

Arbitrary Example

Let $P$ be a Poisson process in which events occur at a rate of $4$ per hour.

Then the mean waiting time of $P$ is $15$ minutes.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponential distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponential distribution