New Mathematical Properties of the Kumaraswamy Lindley Distribution

. The Kumaraswamy Lindley distribution is a generalized distribution that has many applications in various fields, including physics, engineering, and chemistry. This paper introduces new mathematical properties for Kumaraswamy Lindley distribution such as probability weighted moments, moments of residual life, mean of residual life, reversed residual life, cumulative hazard rate function, and mean deviation.

presented Robust reliability estimation for Lindley distribution, a probability integral transform statistical approach, while Hafez et. al [20] presented a study on Lindley distribution accelerated life tests, application, and numerical simulation.

Lindley Distribution and Kumaraswamy Lindley Distribution
The Lindley distribution was introduced in 1958, but it was used as an alternative to the exponential distribution, where the Lindley distribution was used to study many characteristics such as data modeling and other characteristics. In this section, the definition and properties of Lindley distribution are provided. Equation (1) presents the probability distribution function (pdf) of the Lindley distribution with parameter θ: and the corresponding cumulative distribution function (cdf) is Suppose G(x, θ) be the cdf of the Lindley distribution given by (2). The cdf and pdf of Kumaraswamy Lindley distribution (KLD) are Figure 1. Plot of pdf (left) and cdf (right) of KLD

Probability Weighted Moments.
The probabilities weighted moments (PWM) are used to derive parameters of generalized probability distributions and this method is used to compare the parameters obtained by using the probabilities weighted moments: From the definition of the cumulative distribution function of the Kumaraswamy Lindley, we find that Using the binomial theorem on the following and substituting the value of the term into equation number (3) we get Simplifying the equation number, we get Substituting into the cumulative distribution function for a value we get Applying the binomial theorem to the following expression We assume that It becomes probabilities weighted moments (PWM) as following , = 1 + 2 and we calculate the value of each 1 and 2 : The value of probabilities weighted moments (PWM) is , = 1 + 2 , , = ( + + + +2) ( ( + )+ ) + + + +2 The nth moments of residual life denoted by [( − ) | > ] where = 1,2,3, …. It is defined by Substitute in the general form for the residual life for the probability density function of the Kumaraswamy Lindley distribution From the binomial theorem we get Using the binomial expansion of the expression (( − ) ) and the substitution of the general form for residual life, we get We note that ́( ) > 0, > 0, thus mean rof esidual life is increasing (IMRL).

Reversed Residual Life. The nth moments of residual life denoted by [( − ) | ≤ ]
where n = 1, 2, 3, … is defined by We substitute in the general form for the residual life for the probability density function of the Kumaraswamy Lindley distribution, the reversed residual life becomes ( + + + 2, ( + )) ) ( ( + )) + + +2 The reversed of residuals life becomes ( ) = [ ( + + + 1, ( + )) ) ( ( + )) + + +1 + ( + + + 2, ( + )) ) ( ( + )) + + +2 ] The cumulative hazard function (CHF) of the Kumaraswamy Lindley distribution denoted by (x, ) is By substituting for hazard rate function, we get Substituting the value of the probability density function for the Kumaraswamy Lindley distribution as well as the cumulative distribution function for it, we get Applying the rule of integration which states that if the numerator is the differentiation of the denominator, then the integral is Lin (the denominator) Reverse hazard rate function is Substituting for the value of 11, 10 we get

Conclusions
In this research paper, we present some of new mathematical properties of the Kumaraswamy Lindley distribution. The properties of reversed residual life, mean of residual life, moments of residual life, probability weighted moments, and cumulative hazard rate function, have been derived. Furthermore, we invite researchers to study more mathematical properties of the distribution because of its many applications which can contribute to solving many life problems.