# 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 .

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 .

## 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 

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 

References

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

 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.

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

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