In many situations the real data set – from various fields of bio medical, engineering, economical, business etc., cannot identify a best distribution fitting with the conventional distributions, hence a modification or generalisation of the known distribution is significant for the same. There are many methods to modify a distribution. By adding an extra parameter or using the weighted distribution we can formulate one new distribution which satisfies all the properties of probability distribution but its characteristics are completely different from the parent one.
The idea of weighted distributions (WD) plays a significant role in fitting a model to the unknown function of weight while the samples are from the proposed distribution. These distributions provide a remarkable approach that deals with the issue of data interpretation and model specification. Also, it is significant in analysing lifetime data in many subjects like medicine, engineering, finance, and insurance. While the standard distribution does not fit well, the WD is suitable for modelling the statistical data.
The idea of WD, to research how the method of ascertainment can affect the distribution of recorded data was introduced by Fisher^{ 1. }Later this concept is detailed while the common method utilizing the standard distributions was found to be inappropriate executed by Rao^{ 2. }If the weight functions emphasize only the length of units of interest the WD reduces to length biased (LB) distribution. In the context of renewal theory, LB distribution was introduced by Cox^{ 3. }LB sampling situation occurs when a proper sampling frame is absent. LB sampling implies the probability of selecting an element and its magnitude is proportional. LB distribution is the resulting distribution of observations that are selected with probability proportional to their lengths.
A lot of researchers have studied various weighted probability models having examples and applications in various areas. The LB power hazard rate distribution was executed by Mustafa and Khan^{ 4. }AlKadim and Hussein
The ‘2parameter’ Akash distribution (TAD) is a freshly proposed lifetime distribution introduced by Shanker & Shukla
We know that the pdf (probability density function) of TAD is,
The cdf (cumulative distribution function) of TAD is,
The pdf for the weighted random variable
With respect to the many choices of
here
In
and the cdf of P2ND as,
Following equation simplification
Here
From the graphs of pdf (
The structural properties and other characteristics of the P2ND are discussed here.
The
After the simplification of
The initial set of moments of P2ND by letting
Simplifying
By Taylor's series, we obtain,
Also, the characteristic function of P2ND is,
OS plays a key role in statistics and has a wide range of applicability in reliability.
Let
By applying
Then, the pdf of
and the pdf of
Consider the random sample,
To analyze and examine that the random sample comes from the P2ND, the test statistic used is,
We refuse to retain the
Equivalently, we should also refuse to retain the
Whether, if the sample
The BoC and LoC are called classical curves and are being utilized to calculate the distribution of inequality in poverty or income. The BoC & LoC are defined as
and
Simplifying, we get
Entropy plays a key role in various areas of research.
The Renyi Entropy, denoted as e(β). That is,
By binomial expansion,
The Tsallis Entropy for the continuous random variable, it is expressed as,
By binomial expansion in
The MLE (Maximum Likelihood Estimate) of the parameters of P2ND is estimated. For all X_{i}, i=1, 2,…,n. a random sample of the n size by the P2ND, the probability function is,
The solution of these systems of equations by using R program results the MLE of α and θ.
By the asymptotic normality outcomes, attain the CI (Confidence Interval). If
Since
Here, the fitting of a lifetime medical real data in P2NDis considered. It shows that the P2ND fits quite satisfactorily over Lindley, TAD, Akash, and exponential distributions.
The real lifetime medical data (
4.390 
4.395 
4.645 
3.765 
3.750 
3.855 
3.985 
4.050 
0.320 
0.490 
0.620 
1.150 
1.210 
1.260 
1.410 
2.025 
2.910 
3.190 
3.265 
3.350 
3.350 
4.975 
5.075 
5.380 
2.035 
2.160 
2.210 
2.370 
2.530 
2.690 
2.800 
2.910 
2.910 
3.190 
3.265 
3.350 
3.350 
3.430 
3.500 
3.535 
3.765 
3.750 
3.855 
3.985 
4.050 
4.245 
4.325 
4.380 
4.390 
4.395 
4.645 
4.755 
4.930 
4.975 
5.075 
5.380 
3.350 
3.430 
3.500 
3.535 




To compare the performance of P2ND with Lindley, TAD, Akash & exponential distributions, consider the standard general criteria & notations. The lesser values of BIC, AIC, 2logL, and AICC imply the better distribution to which they correspond.










143.7944 
147.7944 
151.1927 
148.1987 



147.2922 
151.2922 
154.67 
151.6165 



152.6894 
154.6894 
156.3783 
154.7946 



170.9553 
172.9556 
174.9245 
172.9661 



159.9501 
161.9501 
163.919 
161.9606 
A generalized format of TAD distribution was suggested and termed asP2ND. Its several statistical properties involving the mean, harmonic mean, variance, moments, BoC, and LoC have been studied. The MLE of the distribution parameters are estimated. P2ND has been examined and investigated with medical data to demonstrate its significance. It is really important to study the characteristics of such biomedical data. The findings show that the suggested P2ND fits across TAD, Akash, exponential, and Lindley distributions rather well.