Research Article Volume 7 Issue 3
Department of Statistics, Eritrea Institute of Technology, Eritrea
Correspondence: Kamlesh Kumar Shukla, Department of Statistics, College of Science, Eritrea Institute of Technology, Asmara, Eritrea
Received: May 10, 2018 | Published: June 21, 2018
Citation: Shukla KK. Pranav distribution with properties and its applications. Biom Biostat Int J. 2018;7(3):244-254. DOI: 10.15406/bbij.2018.07.00215
In this paper, a new one parameter life lime distribution has been proposed and named Pranav distribution. Its statistical and mathematical properties have been derived and its stochastic ordering, mean deviations, Bonferroni and Lorenz curves, order statistics, Renyi entropy measure and stress–strength reliability have been discussed. A Simulation study of proposed distribution has also been discussed. For estimating its parameter method of moments and method of maximum likelihood estimation have been discussed. Goodness of fit of proposed distribution has been presented and compared with other lifetime distributions of one parameter.
Keywords: estimation of parameters, mean deviation, moments, order statistics, reliability measures, renyi entropy measure, stochastic ordering, stress–strength reliability
Many researchers have been proposed many distributions of one parameter as well as two parameters for modeling life time data. Some of them are giving good results for different data sets from biological and engineering which are considered as life time distribution. But some of them are not giving good results for all the data sets from biological and engineering such as Lindley distribution proposed by Lindley1 Akash and Shanker distributions proposed by Shanker2,3 and Ishita distribution proposed by Shanker.4 These distributions are giving good fit over exponential distribution whereas Akash distribution proposed by Shanker2 has been applied on biological data and mentioned the superiority over Lindley distribution and Exponential distribution.
The detailed study about its mathematical properties, estimation of parameter and application has been shown in their paper. Sujatha distribution proposed by Shanker5 has also been applied on real life time data from biological and engineering. Its mathematical and statistical properties has also been discussed in that paper and showed on some selected data set, better than Lindley and Exponential distribution.
Ghitany6 reported in their paper that Lindley distribution is superior to exponential distribution with reference to data related the waiting time before service of the bank customers. In this regard, author interest was to propose new life time distribution which may give good fit in compare to other life time distribution. That is main motivation to propose a new life distribution and applied on biological data.
Therefore, searching a new life time distribution which may be better than Lindley, Exponential, Ishita, Shanker, Sujatha and Akash distribution. It is describe as below:
One parameter life time distribution having parameters θθis defined by its pdf
f(x;θ)=θ4θ4+6(θ+x3)e−θx;x>0,θ>0f(x;θ)=θ4θ4+6(θ+x3)e−θx;x>0,θ>0(1.1)
The pdf (1.1) would call ‘Pranav distribution’ which is a mixture of two–distributions , exponential distribution having scale parameter θθ and gamma distribution having shape parameter 4 and scale parameter, and their mixing proportions of θθ θ4(θ4+6)θ4(θ4+6)and 6(θ4+6)6(θ4+6)respectively.
f2(x;θ)=pg1(x;θ)+(1−p)g2(x;θ,4)f2(x;θ)=pg1(x;θ)+(1−p)g2(x;θ,4)
Where p=θ4(θ4+6),g1(x)=θe−θxp=θ4(θ4+6),g1(x)=θe−θxand g2(x)=θ4x3e−θx6g2(x)=θ4x3e−θx6
The corresponding cumulative distribution function (cdf) of (1.1) is given by
F(x;θ)=1−[1+θx(θ2x2+3θx+6)θ4+6]e−θx;x>0,θ>0F(x;θ)=1−[1+θx(θ2x2+3θx+6)θ4+6]e−θx;x>0,θ>0(1.2)
In this study, new one parameter life time distribution has been proposed and named Pranav distribution. Moment and its related measures have been discussed. Its hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, order statistics, Renyi entropy measure and stress–strength reliability have been discussed. A simulation study of proposed distribution has also been discussed. Both the method of moments and the method of maximum likelihood estimation have been discussed for estimating the parameter of Pranav distribution. The goodness of fit of the Pranav distribution has been presented and the fit has also been compared with other lifetime distributions of one parameter.
Graphs of the pdf and the cdf of Pranav distribution for varying values of parameter are presented in Figure 1 & 2.
The rrth moment about origin of Pranav distribution can be obtained as
μr′=rθr(θ4+6);r=1,2,3,... (2.1)
Thus the first four moments about origin of Pranav distribution are given by
μ1′=θ4+24θ(θ4+6), μ2′=2(θ4+60)θ2(θ4+6),
μ3′=6(θ4+120)θ3(θ4+6),μ4′=24(θ4+210)θ4(θ4+6)
Using above raw moments about origin, central moments of Pranav distribution are thus derived as
μ2=(θ8+84θ4+144)θ2(θ4+6)2
μ3=2(θ12+198θ8+324θ4+864)θ3(θ4+6)3
μ4=9(θ16+312θ12+2304θ8+10368θ4+10368)θ4(θ4+6)4
The coefficient of variation (C.V), coefficient of skewness (√β1), coefficient of kurtosis (β2) and index of dispersion (γ) of Pranav distribution are obtained as
C.V=σμ1′=√(θ8+84θ4+144)(θ4+24)
√β1=μ3μ23/2=2(θ12+198θ8+324θ4+864)(θ8+84θ4+144)3/2
β2=μ4μ22=9(θ16+312θ12+2304θ8+10368θ4+10368)(θ8+84θ4+144)2
γ=σ2μ1′=(θ8+84θ4+144)θ(θ4+6)(θ4+24)
To study the nature of C.V, √β1, β2, and of Pranav distribution, graphs of C.V, √β1, β2, and γ of Pranav distribution have been drawn for varying values of the parameters and presented in Figure 3.
Let X be a continuous random variable with pdf f(x) and cdf F(x). The hazard rate function (also known as the failure rate function) and the mean residual life function of are respectively defined as
h(x)=f(x)1−F(x) (3.1)
m(x)=E[X−x|X>x]=11−F(x)∫∞x[1−F(t)]dt (3.2)
The corresponding hazard rate function, h(x) and the mean residual life function, m(x) of Pranav distribution (1.1) are obtained as
h(x)=θ4(θ+x3)(θ3x3+3θ2x2+6θx+θ4+6) (3.3)
and m(x)=1(θ3x3+3θ2x2+6θx+θ4+6)∞∫x(θ3t3+3θ2t2+6θt+θ4+6)e−θtdt
=(θ3x3+6θ2x2+18θx+θ4+24)θ(θ3x3+3θ2x2+6θx+θ4+6)(3.4)
It can be easily verified that h(0)=θ5θ4+6=f(0) and m(0)=θ4+24θ(θ4+6)=μ1′.
The graph of h(x) and m(x) of Pranav distribution for varying values of parameter is shown in Figure 4&5 respectively.
Stochastic ordering of positive continuous random variables is crucial method for evaluating their comparative behavior. A random variable X is said to be smaller than a random variable Y in the
stochastic order (X≤stY) if FX(x)≥FY(x) for all x
hazard rate order (X≤hrY) if hX(x)≥hY(x) for all x
mean residual life order (X≤mrlY) if mX(x)≤mY(x)for all x
likelihood ratio order (X≤lrY)if fX(x)fY(x)decreases in x.
The following results due to Shaked7 are well known for establishing stochastic ordering of distributions
X≤lrY⇒X≤hrY⇒X≤mrlY
⇓X≤stY
The Pranav distribution is ordered with respect to the strongest ‘likelihood ratio ordering’ as established in the following theorem:
Theorem: Let X and Y follows Pranav distribution with parameters (θ1) and (θ2) respectively. If θ1≥θ2then X≤lrY and hence X≤hrY, X≤mrlY and X≤stY.
Proof: We have
fX(x;θ1)fY(x;θ2)=θ14(θ24+6)θ24(θ14+6)e−(θ1−θ2)x; x>0
Now
lnfX(x;θ1)fY(x;θ2)=ln[θ14(θ24+6)θ24(θ14+6)]−(θ1−θ2)x
This gives ddxlnfX(x;θ1)fY(x;θ2)=−(θ1−θ2)
Thus if θ1>θ2or θ1=θ2, ddxlnfX(x;θ1)fY(x;θ2)<0. This means that X≤lrY and hence X≤hrY, X≤mrlY and X≤stY.
The mean deviation about the mean and median are defined by
δ1(X)=∞∫0|x−μ|f(x)dx and δ2(X)=∞∫0|x−M|f(x)dx, respectively, where μ=E(X)and M=Median (X). The measures δ1(X)and δ2(X)can be calculated using the following simplified relationships
δ1(X)=μ∫0(μ−x)f(x)dx+∞∫μ(x−μ)f(x)dx
=μF(μ)−μ∫0xf(x)dx−μ[1−F(μ)]+∞∫μxf(x)dx
=2μF(μ)−2μ+2∞∫μxf(x)dx
=2μF(μ)−2μ∫0xf(x)dx(5.1)
and
δ2(X)=M∫0(M−x)f(x)dx+∞∫M(x−M)f(x)dx
=MF(M)−M∫0xf(x)dx−M[1−F(M)]+∞∫Mxf(x)dx
=−μ+2∞∫Mxf(x)dx
=μ−2M∫0xf(x)dx(5.2)
Using pdf (1.1) and the mean of Pranav distribution, it can be written as:
μ∫0xf(x;θ)dx=μ−{θ5μ+(μ4+1)θ4+4θ3μ3+12θ2μ2+24θμ+24}e−θμθ(θ4+6)(5.3)
M∫0xf(x;θ)dx=μ−{θ5M+(M4+1)θ4+4θ3M3+12θ2M2+24θM+24}e−θMθ(θ4+6)(5.4)
Using equations (5.1), (5.2), (5.3), and (5.4), the mean deviation about mean, δ1(X)and the mean deviation about median, δ2(X)of Pranav distribution are calculated as
δ1(X)=2{θ4+θ3μ3+6θ2μ2+18θμ+24}e−θμθ(θ4+6)(5.5)
δ2(X)=2{θ5M+(M4+1)θ4+4θ3M3+12θ2M2+24θM+24}e−θMθ(θ4+6)−μ(5.6)
The Bonferroni and Lorenz curves Bonferroni8 and Gini indices have important used in economics to study income and poverty of any state. It’s relevance also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are obtained as
B(p)=1pμq∫0xf(x)dx=1pμ[∞∫0xf(x)dx−∞∫qxf(x)dx]=1pμ[μ−∞∫qxf(x)dx] (6.1)
and L(p)=1μq∫0xf(x)dx=1μ[∞∫0xf(x)dx−∞∫qxf(x)dx]=1μ[μ−∞∫qxf(x)dx](6.2)
respectively or equivalently.
B(p)=1pμp∫0F−1(x)dx(6.3) and L(p)=1μp∫0F−1(x)dx(6.4) respectively, where μ=E(X)and q=F−1(p).
The Bonferroni and Gini indices are thus defined as
B=1−1∫0B(p)dp(6.5) and G=1−21∫0L(p)dp(6.6) respectively.
Using pdf of Pranav distribution (1.1), it can be written
∞∫qxf(x;θ)dx={θ5q+θ4(q4+1)+4θ3q3+12θ2q2+24θq+24}e−θμθ(θ4+6) (6.7)
Now using equation (6.7) in (6.1) and (6.2),
B(p)=1p[1−{θ5q+θ4(q4+1)+4θ3q3+12θ2q2+24θq+24}e−θμ(θ4+6)](6.8)
and L(p)=1−{θ5q+θ4(q4+1)+4θ3q3+12θ2q2+24θq+24}e−θμ(θ4+6)(6.9)
Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini indices of Pranav distribution are thus obtained as
B=1−{θ5q+θ4(q4+1)+4θ3q3+12θ2q2+24θq+24}e−θμ(θ4+6)(6.10)
G=2{θ5q+θ4(q4+1)+4θ3q3+12θ2q2+24θq+24}e−θμ(θ4+6)−1(6.11)
Order statistics
Let X1,X2,...,Xnbe a random sample of size nfrom Pranav distribution (1.1). Let X(1)<X(2)<...<X(n) denote the corresponding order statistics. The pdf and the cdf of the Kth order statistic, say Y=X(k)are given by
fY(y)=n!(k−1)!(n−k)!Fk−1(y){1−F(y)}n−kf(y)
=n!(k−1)!(n−k)!n−k∑l=0(n−kl)(−1)lFk+l−1(y)f(y)
and
FY(y)=n∑j=k(nj)Fj(y){1−F(y)}n−j
=n∑j=kn−j∑l=0(nj)(n−jl)(−1)lFj+l(y)
respectively, for k=1,2,3,...,n.
Thus, the pdf and the cdf of Kth order statistic of Pranav distribution (1.1) are obtained as
fY(y)=n!θ4(θ+x3)e−θx(θ4+6)(k−1)!(n−k)!n−k∑l=0(n−kl)(−1)l ×[1−{θx(θ2x2+3θx+6)+(θ4+6)}e−θx(θ4+6)]k+l−1
and
FY(y)=n∑j=kn−j∑l=0(nj)(n−jl)(−1)l[1−{θx(θ2x2+3θx+6)+(θ4+6)}e−θx(θ4+6)]j+l
Renyi entropy measure
A popular entropy measure is given by Renyi entropy.9 If is a continuous random variable having probability density function f(.), then Renyi entropy is defined as
TR(γ)=11−γlog{∫fγ(x)dx}
where γ>0 and γ≠1.
Thus, the Renyi entropy for Pranav distribution (1.1) can be obtained as
TR(γ)=11−γlog[∞∫0θ4γ(θ4+6)γ(θ+x3)γe−θγxdx]
=11−γlog[∞∫0θ4γ(θ4+6)γθγ(1+x3θ)γe−θγxdx]
=11−γlog[∞∫0θ5γ(θ4+6)γ∞∑j=0(γj)(x3θ)je−θγxdx]
=11−γlog[∞∑j=0(γj)θ5γ(θ2+6)γθj∞∫0e−θγxx3jdx]
=11−γlog[∞∑j=0(γj)θ5γ−j(θ2+6)γ∞∫0e−θγxx3j+1−1dx]
=11−γlog[∞∑j=0(γj)θ5γ−4j−1(θ4+6)γΓ(3j+1)(γ)3j+1]
The stress– strength has wide applications in almost all areas of knowledge especially in engineering such as structures, static fatigue of ceramic components, aging of concrete pressure vessels etc.
LetX and Y be independent strength and stress random variables having Pranav distribution with parameter θ1 and θ2 respectively. Then the stress–strength reliability R can be obtained as
R=P(Y<X)=∞∫0P(Y<X|X=x)fX(x)dx
=∞∫0f(x;θ1,α1)F(x;θ2,α2)dx
=1−θ1[360θ1θ22+1080θ32+144θ2(θ1+θ22)+6(θ1θ32+θ42+6)(θ1+θ23)+6θ1θ22(θ1+θ24)+6θ1θ2(θ1+θ2)5+θ1(θ42+6)(θ1+θ2)6](θ41+6)(θ42+6)(θ1+θ2)7.
Method of Moments Estimates (MOME) of parameters
Method of moments can be calculated equating population mean of Pranav distribution to the sample mean, which is as follows:
MOME of ˜θ θcan be obtained as
θ5ˉx−θ4+6θˉx−24=0(9.1)
Maximum likelihood estimates (mle) of parameters
Let (x1,x2,x3,...,xn) be a random sample of size n from (1.1). The likelihood function, L of Pranav distribution is given by
L=(θ4θ4+6)nn∏i=1(θ+xi3)e−nθˉx
and its log likelihood function is thus obtained as
lnL=nln(θ4θ4+6)+n∑i=1ln(θ+xi3)−nθˉx(9.2)
The maximum likelihood estimates (MLEs) ˆθ of θ,
∂lnL∂θ=4nθ−3nθ(θ4+6)−nˉx=0(9.3)
where ˉx is the sample mean.
The equation (9.3) can be solved directly to estimate the value of θ,
R–Software is used to compute the value of parameter.
A Simulation study has been performed in the present section, it consists in generating N=10,000 pseudo–random samples of sizes 10, 20, 30, 40, 50 from Pranav distribution. Acceptance and rejection method has been used for the simulation of data. Average bias and mean square error of the MLEs of the parameter are estimated using the following formulae
Average Bias = 1Nn∑j=1(⌢θj−θ), MSE=1Nn∑j=1(⌢θj−θ)2
The following algorithm can be used to generate random sample from Pranav distribution. The process to generate a random sample consists of running the algorithm as often as necessary, say ntimes.
Rejection Method for Continuous distribution.10
Suppose Rejection method is used in rectangular area (which can be plot using target density function) to cover the target density fX, then generates candidate points uniformly within the rectangle area. h(x) is candidate density (which is known density function like uniform, exponential, etc.). However if the rectangular area is infinite, then it can cannot generate points uniformly within it, because it has infinite area. Instead it needs a shape with finite area, within which we can simulate points uniformly.
Let X have pdf h and, given X, let Y~U(0,kh(X)) then (X,Y) is uniformly distributed over the region A (say area) defined by the curve kh and 0 are highest and lowest points respectively.
It is noted that the range of Y depends on X.
It can be use conditional probability for example:
P((X,Y)∈(x,x+dx)*(y,y+dy))=P(Y∈(y,y+dy)/X∈(x,x+dx))P(X(x,x+dx))=dykh(x)h(x)dx=1kdxdy
The chance of being in a small rectangle of size dx *dy is the same anywhere in A.
Algorithm
Rejection method: To simulate from the density fx, it is assumed that envelope density h from which it can simulate, and that have some k<∞ such that supxfX(x)h(x)≤k Simulate X from h. where h(x)=θe−θx
The average bias (mean square error) of simulated estimate of parameter θ for different values of n and θ are presented in Table 1.
n |
Parameter θ |
|||
---|---|---|---|---|
0.05 |
0.05 |
1 |
2 |
|
10 |
0.02099(0.004406) |
0.110315(0.121694) |
0.114163(0.130332) |
0.107694(0.11598) |
20 |
0.010726(0.002301) |
0.05524(0.061028) |
0.058333(0.068055) |
0.05615(0.063075) |
30 |
0.006501(0.001268) |
0.034787(0.036304) |
0.036529 (0.04003) |
0.03291(0.032500) |
40 |
0.004479(0.0008025) |
0.024951(0.024903) |
0.025633(0.026284) |
0.021125(0.01785) |
50 |
0.003482(0.0006063) |
0.016715(0.019348) |
0.020210(0.020423) |
0.016287(0.01326) |
Table 1 Average bias (mean square error) of the simulated estimates of parameter θ
The graphs of estimated mean square error of the MLE for different values of parameter θ and nhave been presented in Figure 6.
Data set 1: Present data have been taken from Jones et al.,11 and it is associated with behavioral science. The detailed about data are given in Jones et al.11 A study conducted by the authors in a city located at the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis, which are:
19(16) 20(15) 21(14) 22(9) 23(12) 24(10) 25(6) 26(9) 27(8) 28(5) 29(6) 30(4) 31(3) 32(4) 33 34 35(4) 36(2) 37(2) 39 42 44
Data set 2: This data set is the strength data of glass of the aircraft window reported by:12
18.83 20.8 21.657 23.03 23.23 24.05 24.321 25.5 25.52 25.8 26.69 26.77 26.78 27.05 27.67 29.9 31.11 33.2 33.73 33.76 33.89 34.76 35.75 35.91 36.98 37.08 37.09 39.58 44.045 45.29 45.381
Data Set 3: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm taken from.13
1.312 1.314 1.479 1.552 1.700 1.803 1.861 1.865 1.944 1.958 1.966 1.997 2.006 2.021 2.027 2.055 2.063 2.098 2.140 2.179 2.224 2.240 2.253 2.270 2.272 2.274 2.301 2.301 2.359 2.382 2.382 2.426 2.434 2.435 2.478 2.490 2.511 2.514 2.535 2.554 2.566 2.570 2.586 2.629 2.633 2.642 2.648 2.684 2.697 2.726 2.770 2.773 2.800 2.809 2.818 2.821 2.848 2.880 2.954 3.012 3.067 3.084 3.090 3.096 3.128 3.233 3.433 3.585 3.858
From Above data sets, Pranav distribution has been fitted along with one parameter Sujatha, Ishita and and Akash, Shanker ,Lindley and exponential distributions. pdf, and cdf of these distributions are given in Table 2. The ML estimates, values of−2lnL and K–S statistics of the fitted distributions are shown in Table 3.
Distribution |
cdf |
|
Akash |
f(t)=θ3θ2+2(1+t2)e−θx |
F(t)=1−[1+θt(θt+2)θ2+2]e−θx |
Shanker |
f(t)=θ2θ2+1(θ+t)e−θx |
F(t)=1−[1+θtθ2+1]e−θx |
Sujatha |
f(x;θ)=θ3θ2+θ+2(1+x+x2)e−θx;x>0,θ>0 |
F(x,θ)=1−[1+θx(θx+θ+2)θ2+θ+2]e−θx |
Ishita |
f0(x;θ)=θ3θ3+2(θ+x2)e−θx |
F(x)=1−[1+θx(θx+2)θ3+2]e−θx |
Lindley |
f(x;θ)=θ2θ+1(1+x)e−θx |
F(x;θ)=1−[θ+1+θxθ+1]e−θx |
Exponential |
f(x;θ)=θe−θx |
F(x;θ)=1−e−θx |
Table 2 The pdf and the cdf of fitted distributions
Data Set |
Model |
Parameter estimate |
-2ln L |
AIC |
BIC |
K-S statistic |
Data 1 |
Pranav |
0.160222 |
945.03 |
947.03 |
948.94 |
0.362 |
Ishita |
0.120083 |
980.02 |
982.02 |
984.62 |
0.399 |
|
Sujatha |
0.117456 |
985.69 |
987.69 |
990.29 |
0.403 |
|
Akash |
0.11961 |
981.28 |
983.28 |
986.18 |
0.4 |
|
Shanker |
0.079746 |
1033.1 |
1035.1 |
1037.99 |
0.442 |
|
Lindley |
0.077247 |
1041.64 |
1043.64 |
1046.54 |
0.448 |
|
Exponential |
0.04006 |
1130.26 |
1132.26 |
1135.16 |
0.525 |
|
Data 2 |
Pranav |
0.129818 |
232.77 |
234.77 |
236.68 |
0.253 |
Ishita |
0.097325 |
240.48 |
242.48 |
244.39 |
0.298 |
|
Sujatha |
0.09561 |
241.5 |
243.5 |
245.41 |
0.302 |
|
Akash |
0.097062 |
240.68 |
242.68 |
244.11 |
0.266 |
|
Shanker |
0.064712 |
252.35 |
254.35 |
255.78 |
0.326 |
|
Lindley |
0.062988 |
253.99 |
255.99 |
257.42 |
0.333 |
|
Exponential |
0.032455 |
274.53 |
276.53 |
277.96 |
0.426 |
|
Data 3 |
Pranav |
1.225138 |
217.12 |
219.12 |
221.03 |
0.303 |
Ishita |
0.931571 |
223.14 |
225.14 |
227.05 |
0.33 |
|
Sujatha |
0.936119 |
221.6 |
223.6 |
225.52 |
0.364 |
|
Akash |
0.964726 |
224.28 |
226.28 |
228.51 |
0.348 |
|
Shanker |
0.658029 |
233.01 |
235.01 |
237.24 |
0.355 |
|
Lindley |
0.659 |
238.38 |
240.38 |
242.61 |
0.39 |
|
Exponential |
0.407941 |
261.74 |
263.74 |
265.97 |
0.434 |
Table 3 MLE’s, -2ln L, AIC, BIC, K-S Statistics of the fitted distributions of data-sets 1-3
Profile plots of MLE of parameter and fitted probability plots of proposed distributions, on considered data sets are given in the Figure 7–9 respectively.
In this study, a new one parameter Pranav distribution has been proposed. Its mathematical properties including moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, and stress–strength reliability have been discussed. Simulation study of Pranav distribution has also been discussed. The method of moments and the method of maximum likelihood estimation have been derived for estimating the parameter. Finally, three numerical examples of real lifetime datasets form Biology have been presented to test the goodness of fit of the Pranav distribution. Its fit was found satisfactory over exponential, Lindley, Sujatha, Ishita, Akash and Shanker distributions. It can be considered as good model for biological study.
Note: The paper is named Pranav distribution in the name of my eldest son Pranav Shukla.
None.
Author declares that there is no conflict of interest.
©2018 Shukla. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.
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