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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]

**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.