< List of probability distributions < *Lindley distribution*

## What is the Lindley distribution?

A **Lindley distribution** is one way to describe the lifetime of a process or device. It can be used in a wide variety of fields, including biology, engineering and medicine; it’s especially useful for modeling in mortality studies [1].

The term “Lindley-Exponential Distribution” is often used to mean the generalized form of the distribution.

## Lindley distribution properties

The probability density function (PDF) of a random variable *X* in a Lindley distribution with parameter *θ* is:

The shape parameter (θ) is a positive real number and can result in either a unimodal or monotone decreasing (i.e. consistently decreasing) distribution. The distribution has thin tails because the distribution decreases exponentially for large *x*-values.

The one parameter cumulative distribution function is:

Many forms of the distribution have been described in academic literature, including:

- A two-parameter form [2],
- A two-parameter weighted form [1],
- A generalized Poisson-Lindley [3],
- An extended (EL) distribution [4],
- An exponential geometric distribution [5],
- The transmuted Lindley-Geometric Distribution [6].

Moments for the one-parameter distribution [2]:

**Other parameterizations have different moments**. For example, you can find moments for the Transmuted Lindley-Geometric distribution with the following:

If X has T LG (θ, x) ,φ = (θ, p, λ) then the *r*th moment of X is given by [6]:

Where

And

## Related Distributions

Related distributions that model lifetimes:

**Exponential distribution**: One advantage of the Lindley distribution is that while the exponential distribution has a constant hazard rate and mean residual life function, the Lindley distribution has increasing hazard rate and decreasing mean residual life function. [7]- Gamma distribution.
- Weibull distribution.

## References

[1] Ghitany M. E., Alqallaf F., Al-Mutairi D. K., and Husain H. A., (2011) A two-parameter weighted Lindley distribution and its applications to survival data, Mathematics and Computers in Simulation, Vol. (81), no. 6,1190-1201.

[2] Shanker. On Two – Parameter Lindley Distribution and its

Applications to Model Lifetime Data Biometrics & Biostatistics International Journal Volume 3 Issue 1 – 2016

[3] Mahmoudi E., and Zakerzadeh H., (2010) Generalized Poisson Lindley , Communications in Statistics: Theory and Methods, Vol (39), 1785-1798.

[4] Bakouch H. S., Al-Zahrani B. M., Al-Shomrani A. A., Marchi V. A., and Louzada F.,(2012)

[5] Adamidis K., and Loukas S.,(1998) A lifetime distribution with decreasing failure rate, Statistics and Probability Letters, Vol(39), 35-42.

[6] Merovci, F., and Elbatal, I. (2014) Transmuted Lindley-Geometric Distribution and its Applications. Journal of Statistics Applications and Probability, 3, No. 1, 77-91 (20).

[7] The Lindley Distribution. https://medcraveonline.com/BBIJ/BBIJ-02-00042