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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].

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## Exponential Distribution PDF and CDF

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.

## Linear Exponential Distribution

The **linear exponential (LE) distribution** is an extension of the exponential distribution. It is often used in actuarial science and survival analysis, where it is sometimes called the *linear failure rate distribution*. The LE describes survival patterns with constant initial hazard rates. The “linear” part of the distribution is the hazard rate, which varies as a linear function of age or time [1].

The distribution is one of the best models to fit data that has an increasing failure rate. It is not a good choice for modeling data that has decreasing, non linear increasing, or non-monotone failure rates [2].

## PDF of the Linear Exponential Distribution

If you begin with an exponential distribution with a constant failure rate

f(x) = a + bx,

the result is the linear exponential distribution with a distribution function of [3]

There are, however, a wide range of members in the linear exponential family, so you’ll come across a wide variety of different PDFs, which range from the basic to the complex. For example, the Rayleigh distribution — a submodel of the LE — has PDF

At the other end of the spectrum, the Tweedie distribution — a member of the linear exponential family of distributions, has a PDF that is complex and cannot be expressed in closed form; it’s sometimes expressed as a series of functions.

A two parameter PDF of the linear exponential distribution can be described by [4]

For more details on how the different types are defined, see linear exponential family.

**References**

[1] Lee, E. & Wang, J. (2003). Statistical Methods for Survival Data Analysis. Wiley.

[2] El-Damcese, M. & Marei, Gh. (2012). Extension of the Linear Exponential Distribution and its Applications. International Journal of Science and Research (IJSR). ISSN (Online): 2319-7064.

[3] Mukherjee, S. (2019). A Guide to Research Methodology: An Overview of Research Problems, Tasks and Methods. CRC Press.

[4] Afify, W. (2009). Hyper Linear Exponential Distribution As a Life Distribution. Journal of Applied Sciences Research, 5(12): 2213-2218.

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