Modified Weighted Pareto Distribution Type I (MWPDTI)

: In this paper, the Azzallini’s method used to find a weighted distribution derived from the standard Pareto distribution of type I (SPDTI) by inserting the shape parameter (θ) resulting from the above method to cover the period (0, 1] which was neglected by the standard distribution. Thus, the proposed distribution is a modification to the Pareto distribution of the first type, where the probability of the random variable lies within the period [ 𝑘 𝜃 , ∞) , 𝑘 > 0, 𝜃 ≥ 1. The properties of the modified weighted Pareto distribution of the type I (MWPDTI) as the probability density function ,cumulative distribution function, Reliability function , Moment and the hazard function are found. The behaviour of probability density function for MWPDTI distribution by representing the values of 𝑥 = 𝑘 𝜃 This means, the probability density function of this distribution treats the period (0,1] which is ignore in SPDTI.


Introduction:
The weighted statistical distributions were established in (1934) (1) by Fisher, who pointed to develop and circulate standard statistical distributions to match biased data samples, which are often shown through medical science data and some aspects of biology and other fields of science, Khatree (1989)(1) studied the statistical properties of some weighted length distributions and compared them with the properties of their standard distributions .
The defect in the random sample selection of the population leads to bias in sample size and length, which leads to the failure of standard statistical distributions in the treatment and interpretation of these samples.
In (1978), Patil and Rao (2)presented some concepts on how to use weighted distributions to correct the probability of events in biased samples and apply them to environmental models and human populations. Azzallini (1985)(2) introduced a new technique to find weighted distribution from standard distribution summarized as follows : ( ) = 1 ( 2 < 1 ) Such that ( ) is the probability density function for the weighted distribution, ( 1 )and ( 1 ) are the probability density function and cumulative distribution function of the standard distribution respectively.
Gupt and Kundu (2009)(3) used the Azzallini's method to find the shape parameter for exponential distribution, resulting in a new type of exponential distribution with two parameters. In (2013) (4) Mervat. M. studied the weighted of the Weibull distribution and found its properties Shakhatreh (2012) (5) studied weighted exponential distribution in two properties. Improvement of the Azzallini's method was achieved in (2014) (6)  The standard Pareto distribution type I (SPDTI) has the probability density function ( ) and cumulative distribution function( ) are given respectively by (10): ( ; , ) = +1 , ≥ > 0, > 0 , (2) ( ; , ) = 1 − , ≥ > 0, > 0 (3) The mean and the variance of (PDTI) are respectively given by: Main Result: Azzallini's technique will be used to find weighted distribution derived from Pareto distribution type I, called Modified Weighted Pareto distribution type I (MWPDTI), and study the statistical properties as mean ,variance, standard deviation , cumulative distribution function, Reliability function , hazard rate function, Skewness , Kurtosis and ordered statistics of the MWPDTI. The MWPDTI treated the period (0,1] which was neglected by the SPDTI, such that the value of the random variable is θ , in this treatment the calculate probability values of the random variable by using two parameters k,θ . Thus, the MWPDTI includes the period [ , ∞) , ≥ > 0, ≥ 1.

Weighted Pareto Type I Distribution (WPDTI) Lemma 1:
Let 1 , 2 are a non negative independent random variables having Pareto type I probability density functions (pdf) ( 1 ), ( 2 ) respectively with parameters , , then the is weighted Pareto type I distribution. Proof: Since 1 2 are independent Pareto type I random variables and ≥ , ≥ 1 then By substituting eq. (7) in eq. (1) we get the weighted probability density function (wpdf) of the Pareto distribution type I (WPDTI) is ( ) given by: is a probability density function of the Pareto random variable . Proof: is a weighted probability density function (wpdf).
The purpose of this paper is to expand the standard Pareto distribution of type I by adding the period in which the random variable lies within the , > 0, ≥ > 0, ≥ 1, > 0 is modified weighted probability density function of the Pareto distribution type I. Proof: > , > 0 (10)

Skewness
The skewness of the MWPDTI is . * given by: The order statistics of the MWPDTI Lemma 11:

Conclusion:
The modified weighted distribution of pareto distribution depends on the values of shape parameter obtained from the method of this research which effect directly on the probability density function, survival function and hazard function. In this paper interduce new probability density function contains the shape parameter which given a new weighted distribution for the Pareto distribution type I.
And study the statistical properties for the suggested modified weighted probability density function and other properties which appear by using Azzalini's method to weighted pareto distribution.
The suggested distribution deal with the interval (0, ] which neglects by standard Pareto distribution type I.
The modified weighted standard Pareto distribution type I can be used in a state of the standard Pareto distribution type I, in values for the random variable x less than 1 to near the value of x is 0, otherwise the standard Pareto distribution type I in which the values of x startup the value 1 to ∞.