    # Bivariate Discrete Exponentiated Weibull Distribution: Properties and Applications

In this paper, a new bivariate discrete distribution is introduced which called bivariate discrete exponentiated Weibull (BDEW) distribution. Several of its mathematical statistical properties are derived such as the joint cumulative distribution function, the joint joint hazard rate function, probability mass function, joint moment generating function, mathematical expectation and reliability function for stress-strength model. Further, the parameters of the BDEW distribution are estimated by the maximum likelihood method. Two real data sets are analyzed, and it was found that the BDEW distribution provides better fit than other discrete distributions.

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## 1 Introduction

The Weibull distribution (1951) is one of the most important and well-recognized continuous probability model in research and also in teaching. It has become important because it can be able to assume the properties of many varies types of continuous distributions. And so, if you have right-skewed data, left-skewed data or symmetric data, you can use Weibull to model it. Moreover, the hazard rate of its can be constant, increasing or decreasing. That flexibility of Weibull distribution has made many researchers using it into their data analysis in different fields such as medicine, pharmacy, Engineering, astronomy, electronics, reliability, industry, space science, social sciences, economics and environmental. In previous years, many researchers interested in the distributions theory have provided many generalizations or extensions of the Weibull distribution. See, Mudholkar and Srivastara (1993), Bebbington et al. (2007), Sarhan and Apaloo (2013), El-Gohary et al. (2015), El- Bassiouny et al. (2017), El- Morshedy et al. (2017), among others.

Despite the great importance for the continuous probability distributions, there are many practical cases in which discrete probability distributions are required. Sometimes it is impossible or very difficult to measure the life length of a machine on a continuous scale. For example, on-off switching machines, bulb of photocopier device, etc.

In the last years, many discrete distributions have been derived by discretizing a known continuous distributions. It are obtained by using the same method that used to obtain the discrete geometric (DG) distribution from the continuous exponential distribution. Nakagawa and Osaki (1975) obtained the discrete Weibull (DW) distribution. A new discrete Weibull distribution is proposed by Stein and Dattero (1984). Roy (2003) proposed the discrete normal (DN) distribution. The discrete Rayleigh distribution (DR) is introduced by Roy (2004). Krishna and Pundir (2009) proposed the discrete burr (DB) and discrete Pareto (DP) distributions. Gomez and Calderin (2011) obtained the discrete Lindley (DL) distribution. The discrete generalized exponential (DGE) distribution is proposed by Nekoukhou et al. (2013). Nekoukhou and Bidram (2015) introduced a new three parameters distribution and called it the exponentiated discrete Weibull (EDW) distribution. The CDF and the PMF of the EDW distribution are given respectively by

 FEDW(x;α,p,β)=[1−p([x]+1)α]β,x≥0, (1)

and

 P(X = x)=fEDW(x;α,p,β)=[1−p(x+1)α]β−[1−pxα]β (2) = ∞∑k=1(−1)k+1(βk)[pkxα−pk(x+1)α],x∈N∘={0,1,2,...},

where , and is the largest integer less than or equal . For integer the sum in equation (2) stop at There exist some special discrete distributions can be obtained from EDW distribution as follows:

1. If then the DW distribution of Nakagawa and Osaki (1975) is achieved.

2. If we get the DGE distribution of Nekoukhou et al. (2013).

3. If and then the DG distribution (discrete exponential (DE) distribution) is obtained.

4. If and then the DR distribution of Roy (2004) is achieved.

5. If we get the discrete generalized Rayleigh (DGR) distribution of Alamatsaz et al. (2016).

It is very useful in simulation study for EDW distribution to know the following relation: If

has exponentiated Weibull (EW) distribution, say   then So, to generate a random sample from the EDW distribution, we first generate a random sample from a continuous EW distribution by using the inverse CDF method, and then by considering we find the desired random sample.

On the other hand, the bivariate distributions have been derived and discussed by many authors which have many applications in the areas such as engineering, reliability, sports, weather, drought, among others. Until now, many continuous bivariate distributions based on Marshall and Olkin (1976) model have been introduced in the literature, see Jose et al. (2009), Kundu and Gupta (2009), Sarhan et al. (2011), El-Sherpieny et al. (2013), Wagner and Artur (2013), El- Bassiouny et al. (2016), Rasool and Akbar (2016), El-Gohary et al. (2016), Mohamed et al. (2017), among others.

Also, many discrete bivariate distributions have been introduced, see Kocherlakota and Kocherlakota (1992), Kumar (2008), Kemp (2013), Lee and Cha (2015), Nekoukhou and Kundu (2017), among others.

So, our reasons for introducing the BDEW distribution are the following: to define a bivariate discrete model having different shapes of the hazard rate function, and to define a bivariate discrete model having the flexibility for fitting the real data sets for various phenomena.

The paper is organized as follows: In Section 2, the BDEW distribution is defined. Moreover, the joint CDF and the joint PMF are also presented. Further, some mathematical properties of the BDEW distribution such as the joint PGF, the marginal CDF, the marginal PMF, the conditional PMF of given , the conditional CDF of given the conditional CDF of given , the conditional expectation of given and some other results are presented in Section 3. In Section 4, some reliability studies are introduced. In Section 5, the parameters of the BDEW distribution are estimated by the maximum likelihood method. In Section 6, two real data sets are analyzed to show the importance of the proposed distribution. Finally, Section 7 offers some concluding remarks.

## 2 The BDEW Distribution

Suppose that are three independently distributed random variables, and let . If and

,  then the bivariate vector

has the BDEW distribution with the parameter vector .

##### Lemma 1:

If BDEW(), then the joint CDF of is given by

 FX1,X2(x1,x2) = [1−p(x1+1)α]β1[1−p(x2+1)α]β2[1−p(z+1)α]β3 (3) = ⎧⎪⎨⎪⎩F1(x1,x2) \ \ \ \ \ \ \ % \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ \ x1

where , and are given by

 F1(x1,x2)=FEDW(x1;α,p,β1+β3)FEDW(x2;α,p,β2),
 F2(x1,x2)=FEDW(x1;α,p,β1)FEDW(x2;α,p,β2+β3)

and

 F3(x)=FEDW(x;α,p,β1+β2+β3).
##### Proof:

The joint CDF of the random variables and is defined as follows

 FX1,X2(x1,x2) = P(X1≤x1,X2≤x2) = P(max{V1,V3}≤x1,max{V2,V3}≤x2) = P(V1≤x1,V2≤x2,V3≤min{x1,x2}).

Since, the random variables are independent, we obtain

 FX1,X2(x1,x2) = P(V1≤x1)P(V2≤x2)P(V3≤min{x1,x2}) (4) = FEDW(x1;α,p,β1)FEDW(x2;α,p,β2)FEDW(z;α,p,β3).

By substituting from (1) into (4), we get (3), which complete the proof.

### 2.1 The joint PMF

The joint PMF of the bivariate vector can be easily obtained by using the following relation:

 fX1,X2(x1,x2)=FX1,X2(x1,x2)−FX1,X2(x1−1,x2)−FX1,X2(x1,x2−1)+FX1,X2(x1−1,x2−1). (5)

The joint PMF of for is given by

 fX1,X2(x1,x2)=⎧⎪⎨⎪⎩f1(x1,x2)% \ \ \ \ \ \ \ \ \ \ \ \ \ if \ \ x1

where

 f1(x1,x2) \ = ([1−p(x1+1)α]β1+β3−[1−pxα1]β1+β3)([1−p(x2+1)α]β2−[1−pxα2]β2) = fEDW(x1;α,p,β1+β3)fEDW(x2;α,p,β2),
 f2(x1,x2) = ([1−p(x1+1)α]β1−[1−pxα1]β1)([1−p(x2+1)α]β2+β3−[1−pxα2]β2+β3) = fEDW(x1;α,p,β1)fEDW(x2;α,p,β2+β3)

and

 f3(x) = p1([1−p(x+1)α]β2+β3−[1−pxα]β2+β3)−p2([1−p(x+1)α]β2−[1−pxα]β2) = p1fEDW(x;α,p,β2+β3)−p2fEDW(x;α,p,β2),

where

 p1=[1−p(x+1)α]β1, p2=[1−pxα]β1+β3.

The scatter plot of the joint PMF for the BDEW distribution is given in Figure 1. As expected, the joint PMF for the BDEW distribution can take varies shapes depending on the values of its parameter vector And so, this distribution is more flexible to provide a better fit to variety of data sets.

Figure 1. Scatter plot of the joint PMF of the BDEW distribution for different values of its parameter vector : (a) (b) (c) and (d) .

#### 2.1.1 Special case

Some special bivariate discrete distributions are achieved from the BDEW distribution as follows:

1. If , the bivariate discrete generalized exponential (BDGE) distribution of Nekoukhou and kundu (2017) is obtained.

2. If , we get the bivariate discrete generalized Rayleigh (BDGR) distribution.

3. If and also then we have a new bivariate geomatric (NBG) distribution with two parameters and see, Nekoukhou and kundu (2017). The joint CDF of the NBG distribution is

 FX1,X2(x1,x2)=[1−px1+1]1−β[1−px2+1]1−β[1−pz+1]β,z=min{x1,x2}.

## 3 Statistical Properties

### 3.1 The joint probability generating function

The joint probability generating function (PGF) for any bivariate distribution is very useful and important, because we can use it to find the varies moments, and also the product moments as infinite series. The PGF of the BDEW distribution is mentioned in the following theorem.

##### Theorem 1:

If BDEW(), then the PGF of is given by

 G(u,v) = ∞∑i=0∞∑j=i+1∞∑k=1∞∑l=1(−1)k+l(β1+β3k)(β2l)[pkiα−pk(i+1)α][pkjα−pk(j+1)α]uivj (7) +∞∑j=0∞i=j+1∑∞∑k=1∞l=1∑(−1)k+l(β1k)(β2+β3l)[pkiα−pk(i+1)α][pljα−pl(j+1)α]uivj +∞∑j=0∞∑i=0∞∑k=1(−1)j+k+l(β1k)(β2+β3j)pj(i+1)α[pkiα−pk(i+1)α]uivi −∞∑j=0∞∑i=0∞∑k=1(−1)j+k+l(β1+β3k)(β2j)pjiα[pkiα−pk(i+1)α]uivi,

where and

##### Proof:

From the definition of the joint PGF of we get

 G(u,v)=E(uX1vX2)=∞∑i=0∞∑j=0P(X1=i,X2=j)uivj=I+II+III, (8)

where

 I=∞∑i=0∞∑j=i+1P(X1=i,X2=j)uivj, (9)
 II=∞∑j=0∞∑i=j+1P(X1=i,X2=j)uivj (10)

and

 III=∞∑i=0∞∑j=0P(X1=i,X2=i)uivi. (11)

By substituting from (6) into (9), (10) and (11), we find

 I = ∞∑i=0∞∑j=i+1fEDW(i;α,p,β1+β3)fEDW(j;α,p,β2)uivj = ∞∑i=0∞∑j=i+1∞∑k=1∞∑l=1(−1)k+l(β1+β3k)(β2l)[pkiα−pk(i+1)α][pljα−pl(j+1)α]uivj,
 II = ∞∑j=0∞∑i=j+1fEDW(i;α,p,β1)fEDW(j;α,p,β2+β3)uivj = ∞∑j=0∞∑i=j+1∞∑k=1∞∑l=1(−1)k+l(β1k)(β2+β3l)[pkiα−pk(i+1)α][pljα−pl(j+1)α]uivj

and

 III = ∞∑i=0∞∑j=0[1−p(i+1)α]β1fEDW(i;α,p,β2+β3)uivi (14) −∞∑i=0∞∑j=0[1−piα]β1+β3fEDW(i;α,p,β2)uivi = ∞∑j=0∞∑i=0∞∑k=1(−1)j+k+l(β1k)(β2+β3j)pj(i+1)α[pkiα−pk(i+1)α]uivi −∞∑j=0∞∑i=0∞∑k=1(−1)j+k+l(β1+β3k)(β2j)pjiα[pkiα−pk(i+1)α]uivi.

Substituting from (LABEL:1.19), (LABEL:1.20) and (14) into (8), we get (7), which complete the proof.

### 3.2 The marginal CDF and PMF of X1 and X2

##### Lemma 2:

The marginal CDF of is given by

 FXi(xi)=FEDW(xi;α,p,βi+β3)=[1−p(xi+1)α]βi+β3,xi∈N∘. (15)
##### Proof:

The CDF of given by

 FXi(xi)=P(Xi≤xi)=P(max{Vi,V3}≤xi)=P(Vi≤xi,V3≤xi).

Because the random variables and are independent, we obtain

 FXi(xi) = P(Vi≤xi)P(V3≤xi) = [1−p(xi+1)α]βi[1−p(xi+1)α]β3 = [1−p(xi+1)α]βi+β3=FEDW(xi;α,p,βi+β3).
##### Remark:

The marginal PMF of corresponding to (15) is

 fXi(xi) = fEDW(xi;α,p,βi+β3) (16) = [1−p(xi+1)α]βi+β3−[1−pxαi]βi+β3,xi∈N∘.

### 3.3 The conditional PMF of X1 given X2=x2

The conditional PMF of say is given by

 fX1∣X2=x2(x1∣x2)=⎧⎪⎨⎪⎩f1(x1∣x2) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ \ 0≤x1

where

 f1(x1 ∣ x2)=([1−p(x1+1)α]β1+β3−[1−pxα1]β1+β3)([1−p(x2+1)α]β2−[1−pxα2]β2)[1−p(x2+1)α]β2+β3−[1−pxα2]β2+β3, f2(x1 ∣ x2)=[1−p(x1+1)α]β1−[1−pxα1]β1

and

 f3(x1∣x2)=[1−p(x+1)α]β1−[1−pxα]β1+β3([1−p(x+1)α]β2−[1−pxα]β2)[1−p(x+1)α]β2+β3−[1−pxα]β2+β3.

Equation (17) can be getting by using the following relation:

 fX1∣X2=x2(x1∣x2)=P(X1=x1,X2=x2)P(X2=x2).

### 3.4 The conditional CDF of X1 given X2≤x2

The conditional CDF of say is given by

 FX1∣X2≤x2(x1)=⎧⎪ ⎪⎨⎪ ⎪⎩[1−p(x1+1)α]β1+β3[1−p(x2+1)−β3]\ \ \ \ \ % if \ \ 0≤x1

Equation (18) can be getting by using the following relation

 FX1∣X2≤x2(x1)=P(X1≤x1,X2≤x2)P(X2≤x2).

### 3.5 The conditional CDF of X1 given X2=x2

The conditional CDF of say is given by

 FX1∣X2=x2(x1)=⎧⎪⎨⎪⎩F1(x1∣x2)\ \ \ \ \ \ \ \ \ \ \ \ if \ \ 0≤x1

where

 F1(x1∣x2)=FEDW(x1;α,p,β1+β3 % )fEDW(x2;α,p,β2)fEDW(x2;α,p,β2+β3),
 F2(x1∣x2)=FEDW(x1;α,p,β1)

and

 F3(x1∣x2)=FEDW(x;α,p,β1+β2+β3)− FEDW(x;α,p,β1) FEDW(x−1;α,p,β2+β3 )\ fEDW(x2;α,p,β2+β3).

Equation (19) can be getting by using the following relation

 FX1∣X2=x2(x1)=P(X1≤x1,X2=x2)P(X2=x2),

which

 F1(x1∣x2)=∑j=0x1P(X1=j,X2=x2)P(X2=x2),
 F2(x1∣x2)=∑j=0x1−1P(X1=j,X2=x2)+P(X1=x2,X2=x2)+x2−1∑j=x1+1P(X1=j,X2=x2)P(X2=x2)

and

 F3(x1∣x2)=∑j=0x−1P(X1=j,X2=x2)+P(X1=x,X2=x)P(X2=x2).

### 3.6 The conditional expectation of X1 given X2=x2

##### Lemma 3:

The conditional expectation of say is given by

 E(X1 ∣ X2=x2)=[1−p(x2+1)α]β2−[1−pxα2]β2[1−p(x2+1)α]β2+β3−[1−pxα2]β2+β3 (20) ×∞∑x1=x2+1x1([1−p(x1+1)α]β1+β3−[1−pxα1]β1+β3) +x2−1∑x1=0x1([1−p(x1+1)α]β1−[1−pxα1]β1)+x2[1−p(x2+1)α]β1 −x2[1−pxα2]β1+β3([1−p(x2+1)α]β2−[1−pxα2]β2)[1−p(x2+1)α]β2+β3−[1−pxα2]β2+β3.
 E(X1