Exponential Distribution


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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 [1].  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 [2].

Exponential Distribution PDF and CDF

The exponential distribution PDF.

The general formula for the probability density function PDF) is

Where

  • 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 [2]).

The formula

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

References

[1] Weisstein, Eric W. “Exponential Distribution.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialDistribution.html

[2] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.


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