1 Introduction
The Weibull distribution (1951) is one of the most important and wellrecognized 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 rightskewed data, leftskewed 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), ElGohary 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, onoff 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
(1) 
and
(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:

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

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

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

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

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
the continuous random variable
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), ElSherpieny et al. (2013), Wagner and Artur (2013), El Bassiouny et al. (2016), Rasool and Akbar (2016), ElGohary 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
(3)  
where , and are given by
and
Proof:
The joint CDF of the random variables and is defined as follows
Since, the random variables are independent, we obtain
2.1 The joint PMF
The joint PMF of the bivariate vector can be easily obtained by using the following relation:
(5) 
The joint PMF of for is given by
(6) 
where
and
where
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:

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

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

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
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
(7)  
where and
Proof:
From the definition of the joint PGF of we get
(8) 
where
(9) 
(10) 
and
and
3.2 The marginal CDF and PMF of and
Lemma 2:
The marginal CDF of is given by
(15) 
Proof:
The CDF of given by
Because the random variables and are independent, we obtain
Remark:
The marginal PMF of corresponding to (15) is
(16)  
3.3 The conditional PMF of given
The conditional PMF of say is given by
(17) 
where
and
Equation (17) can be getting by using the following relation:
3.4 The conditional CDF of given
The conditional CDF of say is given by
3.5 The conditional CDF of given
The conditional CDF of say is given by
(19) 
where
and
Equation (19) can be getting by using the following relation
which
and
3.6 The conditional expectation of given
Lemma 3:
The conditional expectation of say is given by
(20)  
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