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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].
Contents:
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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