The exponential distribution, frequently used in reliability tests, describes time between events in a Poisson process, or time between elapsed events. It is a continuous analog of the geometric distribution . The exponential distribution has a wide range of other applications, including in the Monte Carlo method, where random variables from a rectangular distribution are transformed to exponential random variables. Another application is producing approximate solutions for challenging distributional problems .
Exponential Distribution PDF and CDF
The general formula for the probability density function PDF) is
- μ is the location parameter.
- β is the scale parameter.
A variety of other notation is in use. For example, the scale parameter is sometimes also referred to as λ, as shown in the PDF image above, where
λ = 1/β
This process of switching out the two expressions is called reparameterization. One way to think about why we’re using a reciprocal here is to think about what it represents. The reciprocal 1/β is expressed as units of time, while λ is a rate. For example, let’s say you log a sale in your bookstore four times an hour; this is the rate, λ = 4. But you can also express this in units of time: one sale every ¼ of an hour (or 15 minutes).
You might also see the scale parameter as σ [e.g., in ).
Is the PDF for the standard exponential distribution, which has mean (μ) = 0 and scale parameter (β) = 1. This is an example of a one-parameter exponential distribution.
The cumulative distribution function (CDF) is
 Weisstein, Eric W. “Exponential Distribution.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialDistribution.html
 Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.