The Beta-Truncated Lomax Distribution with Communications Data

The beta-truncated Lomax distribution is investigated (BTLD). The goal of this paper is to investigate an extension to Lomax distribution (LD) which is my new distribution BTLD. Also, the LD is a special case of the BTLD. Our distribution can be used as an alternative to the LD. For the BTLD, we obtain the probability density function (PDF), cumulative distribution function (CDF), moments, distribution of order statistics, and a single moment of order statistics. We derive PDF of BTLD minimum order statistics, the PDF of the biggest order statistics. The BTLD's random sum is discussed. An application of the model BTLD to communications real data set is presented to show the superiority of this new distribution by comparing the fitness with LD and exponential distribution. The survival function (SF), hazard function (HF), odd function (OF), reserved hazard function (RHF), moment generating function (MGF), and several moments are all deduced. We get some special cases.


The Beta-Truncated Lomax Distribution
The PDF of the two parametric Lomax distributions (TPLD) is where  and  are the parameters for shape and scale, respectively (Javid and Abdullah, 2015).

If
X has the TPTLD, then we will derive PDF as follows: ( 1) The CDF for equation ( 1) is calculated as: The following three curves:  The PDF of the beta-generated distributions is: By partially differentiating both sides of equation ( 6) with respect to ,b and  equating them to zero, we obtain We cannot obtain the exact solutions for parameters, then the MLEs will be calculated by using Wolfram Mathematica.

Beta-Truncated Lomax Distribution Properties
For BTLD, the PDF and CDF have the The SF is: The HF is: The OF is: The RHF is: The MGF is: where the value  is chosen in such a way that 0. bj

 
Proof: We are well aware of this.The formula for 1 k  is: As a result, the connection can be used to calculate the variance. .

The random Sum for Beta-Truncated Lomax Distribution
Allow the BTLD's PDF to be used The following is how the aforementioned PDF's Laplace transform may be written: follows BTLlD , N is a random variable with a probability mass function (PMF) that is non-negative and integer-valued The following connection gives the Laplace transform of the PDF for the RS of the BTLD: Unit step [t] denotes the unit step function, which has values of 0 if 0  t and 1 if 0  t .

Properties of Order Statistics
In this part we derive some functions and moments for OS.
Let 12 , ,..., n X X X be a random sample from the BTLD of any size n , and let 1: 2: : ...
be the associated OR.The PDF : ,1 rn X r n  is then provided (Javid and Abdullah, 2015)., : ( , ) ( ) ( ) ( ) Using ( 4), ( 5), and ( 13), we get the following PDF of BTLD minimum order statistics: Similarly, the PDF of the biggest order statistics for the BTLD is obtained using (4), (5), and (13): If CDF and PDF of the BTLD are given, then PDF of the OS is given.
We use the BTLD to create explicit expressions for OS moments.
is the associated OS.The the value  is selected in a way Permit me to elaborate.
As a result, (6.3) becomes   1 () :: 00 We'll offer some cases: When this 1 k  is combined with relation ( 16), we get the following:   : : : 16), we may get the th r OS's second-order moment as   (2) 2 2 : : : As a result, the th r order statistic's variance can be calculated using the relationship.
  The smallest order statistic's secondorder moment can also be calculated as: As a result, the greatest order statistic's mean, second-order moment, and variance can be calculated as order statistics (OS) for the two parametric Lomax distribution (LD).The exponential LD is obtained by El-Bassiouny et al., 2015.In Wake et al. 2003, they derived the form of the probability density function (PDF) from the evaluation in time of a previously truncated frequency distribution of animal live weights.Plenty of problems can be solved by reference to the random sum (RS). ) is considered, we derive the PDF and the CDF of BTLD in Section 1.In addition, we exhibit the PDF of the BTLD for various parameter values.In Section 2, an application of communications real data set is presented.The SF, HF, OF, RHF, MGF, and several moments are all deduced in Section 3. The BTLD's RS is obtained in Section 4. Section 5 includes some distributions and moments order statistics.In Sections 6, we obtain some special cases.
the PDF for TPTLD.

Fig
Fig. (1): The relationship between  and the PDF of TPTLD the BTLD's PDF and CDF are determined using the relation (3).
Fig. (2): The relationship between  and the PDF of BTLD From Fig. (2), when the parameter values , b  ) the MLEs for the parameters of BTLD (  , α, b), LD (  , α) and exponential distribution (ED) are obtained.The value of the negative log likelihood function (NLOG), Bayesian information criterion (BIC), Akaike information criterion (AIC), Hannan-Quinn information criterion (HQIC) and second order of Akaike information criterion (AICc) are evaluated in table