Probability Distributions > Lindley Distribution

## What is a 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. Ghitany et. al (2011) state that it’s especially useful for modeling in mortality studies. The shape parameter, θ is a positive real number and can result in either a unimodal or monotone decreasing (i.e. consistently decreasing) distribution. This distribution has thin tails because the distribution decreases exponentially for large x-values. The term “Lindley-Exponential Distribution” is often used to mean the generalized form of the distribution.

## PDF and CDF

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

The cumulative distribution function for the one-parameter function is:

## Types

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

- A two-parameter form (Shanker et. al),
- A two-parameter weighted form (Ghitany et’ al),
- A generalized Poisson-Lindley (Mahmoudi et. al.),
- An extended (EL) distribution (Bakouch et. al),
- An exponential geometric distribution (Adankis and Loukas).
- The transmuted Lindley-Geometric Distribution (Merovci, and Elbatal)

## Moments

Moments for the one-parameter distribution (from Shanker et. al):

Note that these moments are only valid for the one-parameter form. **Other parameters will 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 rth moment of X is given by (from Merovci and Elbatal):

Where

and

## Related Distributions

Related distributions that model lifetimes:

**Exponential Distribution**. According to Shanker et. al (1995), one advantage of the Lindely distribution compared to the exponential distribution is that “…the exponential distribution has constant hazard rate and mean residual life function whereas the Lindley distribution has increasing hazard rate and decreasing mean residual life function.”**Gamma Distribution**,**Weibull distribution**.

**References**:

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

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

An extended LD, Journal of the Korean Statistical Society, Vol(41), 75-85.

Bhati, D., and Malik, A.,(2014), On Lindley-Exponential Distribution: Properties and Application. Retrieved 8/24/2016 from https://arxiv.org/pdf/1406.3106.pdf.

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.

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

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). Retrieved 8/24/2016 from http://naturalspublishing.com/files/published/e69vpy514e5z24.pdf.

Shanker R., Sharma S., and Shanker R., (2013) A Two-Parameter Lindley for

Modeling Waiting and Survival Times Data, Applied Mathematics, Vol (4), 363-368. Retrieved 8/24/2016 from here.

Shanker et. al (2015). On Modeling of Lifetimes Data Using Exponential and Lindley Distributions. Biometrics & Biostatistics International Journal. Volume 2 Issue 5.