Exponential distribution
The exponential distribution models the waiting time until the next event when events occur at rate lambda. P(X > t) = e^(-lambda t). The mean is 1/lambda. It is the waiting-time partner of the Poisson.
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Analogy
How long until the next shooting star, if they arrive randomly at a steady average rate.
Example
Rate lambda = 0.5 per minuteMean wait = 1 / 0.5 = 2 minP(wait > 3) = e^(-1.5) = 0.223P(wait <= 3) = 1 - 0.223 = 0.777
In practice
Used for waiting times between customers, and for the lifetime of parts that do not wear out.
Practice
Flashcards
- Mean of an exponential distribution with rate lambda?1 / lambda.
Questions people ask
How are the exponential and Poisson distributions related?
If events occur at random with rate lambda, the number in a fixed time is Poisson, and the gap between consecutive events is exponential. One counts events, the other measures the wait. Both describe the same underlying random process, seen from two different angles, so they share the same rate lambda.
Why is the exponential distribution called memoryless?
If you have already waited 5 minutes, the extra time you still must wait has the same distribution as a fresh wait. The past gives no information. That suits events with a constant rate, but not parts that wear out with age.