Exponential-logarithmic distribution

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The Exponential-Logarithmic distribution is used to model decreasing failure rates. This log-series mixture distribution, introduced by Tahmasbi and Rezaei in 2008 [1], is created when the rate parameter for the exponential distribution is randomized by the logarithmic distribution. In this blog, we will discuss why and how Exponential-logarithmic (EL) distributions are used, and delve into their properties and applications.

What is an Exponential-logarithmic distribution?

The Exponential-Logarithmic (EL) distribution is a family of lifetime distributions with decreasing failure rate, defined on the interval [0, ∞). An EL distribution can be thought of as a mixture of two different probability distributions—an exponential random variable and a logarithmic random variable. This mixture creates a new probability distribution that is especially useful in modeling decreasing failure rates over time. For example, it can be used to calculate the “time-to-failure” for electronic components or other machinery that degrades with use.

EL Distribution Properties

PDF of the exponential-logarithm distribution, with p = 0.5 and β = 2 (red) and 3 (blue).

The probability density function (PDF) is usually expressed as [1]:

exponential-logarithm distribution formula
p ∈ (0, 1) and β > 0.

This function is strictly decreasing in x and tends to zero as x tends to infinity (x →

The cumulative distribution function (CDF) is:

el distribution cdf

The mode of the EL distribution is zero, given by

(β(1 – p)) / –p ln p).

And the median is given by

(ln(1 + √p))/ β

The properties of an EL distribution depend on its two components, the exponential random variable and the logarithmic random variable. The exponential part of the EL equation describes probabilities that decrease exponentially over time, while the logarithmic part indicates probabilities that decrease more slowly or remain constant over time. This combination of components makes it possible to accurately model decreasing failure rates in many situations.

A key feature of an EL distribution is its shape—its peaked shape can help us identify trends or patterns in data sets related to decreasing failure rates. Additionally, its asymmetric shape allows us to predict values outside our range of observations more accurately than other probability distributions like normal distributions might allow us to do.


Graph created with Desmos.

[1] Tahmasbi, R. and Rezaei, S. (2008). A two-parameter lifetime distribution with decreasing failure rate. Computational Statistics & Data Analysis 52(8):3889-3901

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