In this section, we introduce two new families of bivariate distributions with marginals having distributions with forms (1) or (2). We apply a similar technique of that proposed by Marshall and Olkin [15], for obtaining these families.

### The construction of the families (models)

#### Lifetime model

Suppose that a system consists of two subsystems, say A and *B*. Subsystem A contains two components, say A_{1}, and *C*, connected in series (parallel) with lifetimes U_{1} and *U*_{0} , respectively. Subsystem *B* contains the two components, say *B*_{1} and *C*, connected in series (parallel), where the lifetime of component *B*_{1} is *U*_{2}.

Suppose that *U*_{i}, *i* = 0, 1, 2. , are independent random variables following EF(θ_{i}, c), *i* = 0, 1, 2 for the series case and IEF(*β*_{i}; c), *i* = 0, 1, 2. , for the parallel case, i.e.,

$$ {\overline{F}}_{U_i}(u)=\left\{\begin{array}{c}{\overline{F}}_{\mathrm{EF}}\left(u;\theta, c\right)={e}^{-{\theta}_i{g}_1\left(u;c\right)},i=0,1,2,\mathrm{for}\ \mathrm{the}\ \mathrm{series}\ \mathrm{case},\\ {}{\overline{F}}_{\mathrm{IEF}}\left(u;\beta, c\right)=1-{e}^{-{\beta}_i{g}_2\left(u;c\right)},i=0,1,2,\mathrm{for}\ \mathrm{the}\ \mathrm{parallel}\ \mathrm{case}.\end{array}\right. $$

(3)

If *X* and *Y* are the lifetimes of the two subsystems *A* and *B*, respectively, then we have *X* = min {*U*_{0}, *U*_{1}} and *Y* = min {*U*_{0}, *U*_{2}}., for the series case, while *X* = max {*U*_{0}, *U*_{1}} and *Y* = max {*U*_{0}, *U*_{2}}, for the parallel case.

#### Stress model

Consider a two-component system and consider three independent stresses say *U*_{0}, *U*_{1}, and *U*_{2}. Each component is subject to an individual stress say *U*_{1} and *U*_{2}, respectively, while *U*_{0} is an overall stress transmitted to both the components equally. Then,

- 1.
The observed stress on the two components is *X* = max {*U*_{0}, *U*_{1}} and *Y* = max {*U*_{0}, *U*_{2}}., respectively.

- 2.
If the stresses are always fatal, then the lifetime of the two components are *X* = min {*U*_{0}, *U*_{1}} and *Y* = min {*U*_{0}, *U*_{2}}.

We can observe that in the two models there is the possibility of having *X* = *Y*; thus, the two models have both an absolute continuous part and a singular part, similar to M-O bivariate exponential model.

Theorems 1–3 present the survival functions and the probability density functions of the proposed bivariate families.

**Theorem 1** Suppose *U*_{i}, *i* = 0, 1, 2., are independent random variables following *EF*(*θ*_{i}; *c*), *i* = 0, 1, 2., and let *X* = min {*U*_{0}, *U*_{1}} and *Y* = min {*U*_{0}, *U*_{2}}; then, the bivariate vector (*X*, *Y*) will have the survival function

$$ {\overline{F}}_{BEF}\left(X,Y\right)=\exp \left\{-{\theta}_1{g}_1\left(x;c\right)-{\theta}_2{g}_1\left(y;c\right)-{\theta}_0{g}_1\left(\max \left\{x,y\right\};c\right)\right\}. $$

(4)

**Proof** Obviously, from \( {\overline{F}}_{X,Y}\left(x,y\right)=P\left(X>x,Y>y\right) \), we can write \( {\overline{F}}_{\mathrm{BEF}}\left(X,Y\right) \) as

$$ P\left(\min \left\{{U}_0,{U}_1\right\}>x,\min \left\{{U}_0,{U}_2\right\}>y\right)=P\left({U}_1>x,{U}_2>y,{U}_0>\min \left(x,y\right)\right). $$

Since *U*_{i} are independent random variables following *EF*(*θ*_{i}; *c*), *i* = 0, 1, 2. Hence, (4) holds. ∎

We will denote the bivariate distribution with survival function having the form (4) by BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*). Clearly, *X* and *Y* are independent if and only if (iff) *θ*_{0} = 0. The joint survival function can also written as

$$ {\overline{F}}_{\mathrm{BEF}}\left(X,Y\right)=\left\{\begin{array}{c}\exp \left\{-\left({\theta}_0+{\theta}_1\right){g}_1\left(x;c\right)-{\theta}_2{g}_1\left(y;c\right)\right\},\kern0.75em \mathrm{if}\ x\ge y\\ {}\exp \left\{-{\theta}_1{g}_1\left(x;c\right)-\left({\theta}_0+{\theta}_2\right){g}_1\left(y;c\right)\right\},\kern0.5em \mathrm{if}\ y>x\ \end{array}\right. $$

**Theorem 2** Suppose *U*_{i}, *i* = 0, 1, 2. , are independent random variables following *EIF*(*β*_{i}; *c*), *i* = 0, 1, 2. , and let *X* = max {*U*_{0}, *U*_{1}} and *Y* = max {*U*_{0}, *U*_{2}}; then, the bivariate vector (*X*, *Y*) has the cumulative function

$$ {F}_{\mathrm{BIEF}}\left(X,Y\right)=\exp \left\{-{\beta}_1{g}_2\left(x;c\right)-{\beta}_2{g}_2\left(y;c\right)-{\beta}_0{g}_2\left(\min \left\{x,y\right\};c\right)\right\}. $$

(5)

**Proof** Similarly as in Theorem 1, using *F*_{X, Y}(*x*, *y*) = *P*(*X* < *x*, *Y* < *y*), we can show that (5) holds. ∎

We will denote the bivariate distribution with cumulative function with the form (5) by BIEF(β_{0}, β_{1}, β_{2}; c). Clearly, *X* and *Y* are independent iff *β*_{0} = 0. The joint cumulative function can also be written as

$$ {F}_{\mathrm{BIEF}}\left(X,Y\right)=\left\{\begin{array}{c}\exp \left\{-{\beta}_1{g}_2\left(x;c\right)-\left({\beta}_0+{\beta}_2\right){g}_2\left(y;c\right)\right\},\mathrm{if}\ x\ge y\\ {}\exp \left\{-\left({\beta}_0+{\beta}_1\right){g}_2\left(x;c\right)-{\beta}_2{g}_2\left(y;c\right)\right\},\mathrm{if}\ y>x\ \end{array}\right. $$

**Theorem 3** If the vector (*X*, *Y*) has either BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*) or BIEF(β_{0}, β_{1}, β_{2}; c), then their joint pdf is given by

$$ {f}_{X,Y}\left(x,y\right)=\left\{\begin{array}{c}{f}_1\left(x,y\right),\mathrm{if}\ x>\mathrm{y}\\ {}{f}_2\left(x,y\right),\mathrm{if}\ x<y\\ {}{f}_0(x),\mathrm{if}\ x=y\end{array}\right. $$

(6)

where \( {\displaystyle \begin{array}{c}{f}_1\left(x,y\right)=\left\{\begin{array}{c}{\theta}_2\left({\theta}_0+{\theta}_1\right){g}_1^{\prime}\left(x;c\right){g}_1^{\prime}\left(y;c\right)\ {\mathrm{e}}^{-\left({\theta}_0+{\theta}_1\right){g}_1\left(x;c\right)-{\theta}_2{g}_1\left(y;c\right)},\mathrm{for}\ \mathrm{BEF}\left({\theta}_0,{\theta}_1,{\theta}_2;c\right)\\ {}\ {\beta}_1\left({\beta}_0+{\beta}_2\right){g}_2^{\prime}\left(x;c\right){g}_2^{\prime}\left(y;c\right){e}^{-{\beta}_1{g}_2\left(x;c\right)-\left({\beta}_0+{\beta}_2\right){g}_2\left(y;c\right)},\mathrm{for}\ \mathrm{BIEF}\left({\upbeta}_0,{\upbeta}_1,{\upbeta}_2;\mathrm{c}\right)\end{array}\right.\\ {}{f}_2\left(x,y\right)=\left\{\begin{array}{c}{\theta}_1\left({\theta}_0+{\theta}_2\right){g}_1^{\prime}\left(x;c\right){g}_1^{\prime}\left(y;c\right)\ {\mathrm{e}}^{-{\theta}_1{g}_1\left(x;c\right)-\left({\theta}_0+{\theta}_2\right){g}_1\left(y;c\right)},\mathrm{for}\ \mathrm{BEF}\left({\theta}_0,{\theta}_1,{\theta}_2;c\right)\\ {}\ {\beta}_2\left({\beta}_0+{\beta}_1\right){g}_2^{\prime}\left(x;c\right){g}_2^{\prime}\left(y;c\right){e}^{-\left({\beta}_0+{\beta}_1\right){g}_2\left(x;c\right)-{\beta}_2{g}_2\left(y;c\right)},\mathrm{for}\ \mathrm{BIEF}\left({\upbeta}_0,{\upbeta}_1,{\upbeta}_2;\mathrm{c}\right)\end{array}\right.\end{array}} \)

and

$$ {f}_0(x)=\left\{\begin{array}{c}{\theta}_0{g}_1^{\prime}\left(x;c\right){\mathrm{e}}^{-\theta {g}_1\left(x;c\right)},\mathrm{for}\ \mathrm{BEF}\left({\theta}_0,{\theta}_1,{\theta}_2;c\right)\\ {}-{\beta}_0{g}_2^{\prime}\left(x;c\right){e}^{-\beta {g}_2\left(x;c\right)\Big)},\mathrm{for}\ \mathrm{BIEF}\left({\upbeta}_0,{\upbeta}_1,{\upbeta}_2;\mathrm{c}\right)\end{array}\right. $$

With *θ* = *θ*_{0} + *θ*_{1} + *θ*_{2}, *β* = *β*_{0} + *β*_{1} + *β*_{2} and \( {g}_i^{\prime}\left(t;c\right),i=1,2, \) is the first derivative of *g*_{i}(*t*; *c*) with respect to *t*.

**Proof** Clearly, for the two models, *f*_{1}(*x*, *y*) and *f*_{2}(*x*, *y*) can be easily obtained by using \( \frac{\partial^2{\overline{F}}_{X,Y}\left(x,y\right)}{\partial x\partial y} \) or \( \frac{\partial^2{F}_{X,Y}\left(x,y\right)}{\partial x\partial y} \) for *x* > *y* and *y* > *x* respectively. For *f*_{0}(*x*), we use the relation

\( {\int}_0^{\infty }{\int}_0^x{f}_1\left(x,y\right) dydx+{\int}_0^{\infty }{\int}_0^y{f}_2\left(x,y\right) dx dy+{\int}_0^{\infty }{f}_0(x) dx=1 \). So, for the BEF, we have

$$ {\int}_0^{\infty }{\int}_0^x{f}_1\left(x,y\right) dydx=1-\left({\theta}_0+{\theta}_1\right){\int}_0^{\infty }{g}_1^{\prime}\left(t;c\right){\mathrm{e}}^{-\theta {g}_1\left(t;c\right)} dt $$

and

$$ {\int}_0^{\infty }{\int}_0^y{f}_2\left(x,y\right) dxdy=1-\left({\theta}_0+{\theta}_2\right){\int}_0^{\infty }{g}_1^{\prime}\left(t;c\right){\mathrm{e}}^{-\theta {g}_1\left(t;c\right)} dt, $$

Thus,

$$ {\int}_0^{\infty }{f}_0(x) dx=1-\left[2-\left({\theta}_0+\theta \right){\int}_0^{\infty }{g}_1^{\prime}\left(t;c\right){\mathrm{e}}^{-\theta {g}_1\left(t;c\right)} dt\right]={\theta}_0{\int}_0^{\infty }{g}_1^{\prime}\left(t;c\right){\mathrm{e}}^{-\theta {g}_1\left(t;c\right)} dt. $$

Similarly for the BIEF, we have \( {\int}_0^{\infty }{f}_0(x) dx=1+\left({\beta}_1+{\beta}_2\right){\int}_0^{\infty }{g}_2^{\prime}\left(t;c\right){\mathrm{e}}^{-\beta {g}_2\left(t;c\right)} dt=-{\beta}_0{\int}_0^{\infty }{g}_2^{\prime}\left(t;c\right){\mathrm{e}}^{-\beta {g}_2\left(t;c\right)} dt \).

Hence, the proof is complete. ∎

Notice that both distribution BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*) and BIEF (*β*_{0}, *β*_{1}, *β*_{2}; *c*) are singular on the line *X* = *Y*, since *P*(*X* = *Y*) ≠ 0. Thus the two models have a singular part and an absolute continuous part, similar to Marshall and Olkin’s model. The following theorem provides explicitly the absolute continuous part and the singular part of BEF and BIEF.

**Theorem 4** If the vector (*X*, *Y*) has BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*) or BIEF(*β*_{0}, *β*_{1}, *β*_{2}; *c*), then

- (i)
The survival function for the BEF is

$$ {\overline{F}}_{\mathrm{BEF}}\left(x,y\right)=\frac{\theta_1+{\theta}_2}{\theta }{\overline{F}}_{\mathrm{BEF}(a)}\left(x,y\right)+\frac{\theta_0}{\theta }{\overline{F}}_{\mathrm{BEF}(s)}\left(x,y\right), $$

(7)

Where, *θ* = *θ*_{0} + *θ*_{1} + *θ*_{2}, \( {\overline{F}}_{\mathrm{BEF}(s)}\left(x,y\right)={\mathrm{e}}^{-\theta {g}_1\left(\max \left\{x,y\right\};c\right)} \) is the singular part, and \( {\overline{F}}_{\mathrm{BEF}(a)}\left(x,y\right)=\frac{\theta }{\theta_1+{\theta}_2}{\mathrm{e}}^{-{\theta}_1{g}_1\left(x;c\right)-{\theta}_2{g}_1\left(y;c\right)-{\theta}_0{g}_1\left(\max \left\{x,y\right\};c\right)}-\frac{\theta_0}{\theta_1+{\theta}_2}{\mathrm{e}}^{-\theta {g}_1\left(\max \left\{x,y\right\};c\right)} \) is the absolute continuous part.

- (ii)
The cumulative function for the BIEF is

$$ {F}_{\mathrm{BIEF}}\left(x,y\right)=\frac{\beta_1+{\beta}_2}{\beta }{F}_{\mathrm{BIEF}(a)}\left(x,y\right)+\frac{\beta_0}{\beta }{F}_{\mathrm{BIEF}(s)}\left(x,y\right), $$

(8)

where, *β = β*_{0} + *β*_{1} + *β*_{2}, \( {F}_{\mathrm{BIEF}(s)}\left(x,y\right)={\mathrm{e}}^{-\beta {g}_2\left(\min \left\{x,y\right\};c\right)} \) is the singular part and \( {F}_{\mathrm{BIEF}(a)}\left(x,y\right)=\frac{\beta }{\beta_1+{\beta}_2}{\mathrm{e}}^{-{\beta}_1{g}_2\left(x;c\right)-{\beta}_2{g}_2\left(y;c\right)-{\beta}_0{g}_2\left(\min \left\{x,y\right\};c\right)}-\frac{\beta_0}{\beta_1+{\beta}_2}{\mathrm{e}}^{-\beta {g}_2\left(\min \left\{x,y\right\};c\right)} \) is the absolute continuous part.

**Proof** (i) For the BEF, using the fact that \( {\overline{F}}_{\mathrm{BEF}}\left(x,y\right)=\alpha {\overline{F}}_{\mathrm{BEF}(a)}\left(x,y\right)+\left(1-\alpha \right){\overline{F}}_{\mathrm{BEF}(s)}\left(x,y\right) \)

$$ \frac{\partial^2{\overline{F}}_{\mathrm{BEF}}\left(x,y\right)}{\partial x\partial y}=\alpha {f}_{\mathrm{BEF}(a)}\left(x,y\right)=\left\{\begin{array}{c}{f}_{\mathrm{EF}}\left(x;{\theta}_0+{\theta}_1,c\right){f}_{\mathrm{EF}}\left(y;{\theta}_2,c\right),\mathrm{if}\ x>y\\ {}{f}_{\mathrm{EF}}\left(x;{\theta}_1,c\right){f}_{\mathrm{EF}}\left(y;{\theta}_0+{\theta}_2,c\right),\mathrm{if}\ x<y\end{array}\right. $$

Hence *α* may be obtained as

$$ \alpha =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{x}{\int }}{f}_{\mathrm{EF}}\left(x;{\theta}_0+{\theta}_1,c\right){f}_{\mathrm{EF}}\left(y;{\theta}_2,c\right) dydx+\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{y}{\int }}{f}_{\mathrm{EF}}\left(x;{\theta}_1,c\right){f}_{\mathrm{EF}}\left(y;{\theta}_0+{\theta}_2,c\right) dxdy=\frac{\theta_1+{\theta}_2}{\theta }, $$

and \( {\overline{F}}_{\mathrm{BEF}(a)}\left(x,y\right)=\underset{y}{\overset{\infty }{\int }}\underset{x}{\overset{\infty }{\int }}{f}_{\mathrm{BEF}(a)}\left(u,v\right) dudv \); hence, with α and \( {\overline{F}}_{\mathrm{BEF}(a)}\left(x,y\right) \) known, the singular part \( {\overline{F}}_{\mathrm{BEF}(s)}\left(x,y\right) \) can be obtained by subtraction.

(ii) Similarly for the BIEF, *F*_{BIEF(a)}(*x*, *y*) is computed by using *F*_{BIEF}(*x*, *y*) = *γF*_{BIEF(a)}(*x*, *y*) +(1 − *γ*)*F*_{BIEF(s)}(*x*, *y*), 0 ≤ *γ* ≤ 1. Using a similar manner as in part (i), we can show that (8) holds. ∎

The marginal distributions of X and Y and the conditional distributions are given by Theorems 5 and 6, while the distributions of min{*X*, *Y*}, for the BEF, and max{*X*, *Y*}, for the BIEF, are given by Theorem 7.

**Theorem** 5 If the vector (*X*, *Y*) has either BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*) or BIEF(*β*_{0}, *β*_{1}, *β*_{2}; *c*), then the marginal distributions of X and Y are either EF(*θ*_{0}, *θ*_{i}; *c*) or IEF (*β*_{0}, *β*_{i}; *c*), *i=*1,2, respectively.

**Proof** If (*X*, *Y*) has BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*), then from (6) we have

$$ {f}_X(x)=\underset{0}{\overset{x}{\int }}{f}_1\left(x,y\right) dy+\underset{x}{\overset{\infty }{\int }}{f}_2\left(x,y\right) dy+{f}_0(x)=\left({\theta}_0+{\theta}_1\right){g}_1^{\prime}\left(x;c\right)\ {\mathrm{e}}^{-\left({\theta}_0+{\theta}_1\right){g}_1\left(x;c\right)}. $$

Similarly we can derive *f*_{Y}(*y*). In a similar manner, *f*_{X}(*x*) and *f*_{Y}(*y*) can be shown to have IEF (*β*_{0}, *β*_{i}; *c*), *i=*1,2, respectively for the BIEF. ∎

Notice that the marginal distributions of X and Y can also be obtained using the next lemma.

**Lemma 1**

(i) Let *X* = min {*U*_{0}, *U*_{1}}, then *X* ∼ *EF*(*θ*_{0} + *θ*_{1}; *c*) iff *U*_{0} and *U*_{1} are independent and *U*_{0} ∼ *EF*(*θ*_{0}; *c*), *U*_{1} ∼ *EF*(*θ*_{1}; *c*).

(ii) Let *X* = max {*U*_{0}, *U*_{1}}, then *X* ∼ *IEF*(*β*_{0} + *β*_{1}; *c*) iff *U*_{0} and *U*_{1} are independent and *U*_{0} ∼ *IEF*(*β*_{0}; *c*), *U*_{1} ∼ *IEF*(*β*_{1}; *c*).

Here “∼” means follows or has the distribution.

**Proof** (i) for *X* = min {*U*_{0}, *U*_{1}}, we have

$$ P\left(X>x\right)=P\left(\min \left\{{U}_0,{U}_1\right\}>x\right)=P\left({U}_0>x,{U}_1>x\right). $$

If *U*_{0} and *U*_{1} are independent and *U*_{0} ~ EF(*θ*_{0}; *c*) and *U*_{1} ~ EF(*θ*_{1}; *c*) *U*_{1} ∼ EF(*θ*_{1}; *c*), then

$$ P\left(X>x\right)=P\left({U}_0>x\right)P\left({U}_1>x\right)={e}^{-\left({\theta}_0+{\theta}_1\right){g}_1\left(x;c\right)}. $$

Conversely, if *X* ∼ *EF*(*θ*_{0} + *θ*_{1}; *c*), then

$$ P\left(X>x\right)={e}^{-\left({\theta}_0+{\theta}_1\right){g}_1\left(x;c\right)}={e}^{-{\theta}_0{g}_1\left(x;c\right)}{e}^{-{\theta}_1{g}_1\left(x;c\right)}. $$

Then, *U*_{0} and *U*_{1} are independent and \( {\overline{F}}_{U_0}(x)={e}^{-{\theta}_0{g}_1\left(x;c\right)} \) and \( {\overline{F}}_{U_1}(x)={e}^{-{\theta}_1{g}_1\left(x;c\right)}, \) i.e. U_{0} ∼ EF(θ_{0}; c) and U_{1} ∼ EF(θ_{1}; c).

- (ii)
Similarly for the BIEF. ∎

Consequently, from Theorems 1 and 2 and Lemma 1, we have the following lemma, Lemma 2.

**Lemma 2**

(i) (*X*, *Y*) ∼ BEF(θ_{0}, θ_{1}, θ_{2}; c) iff there exist independent EF random variables *U*_{i}, *i* = 0, 1, 2, such that *X* = min {*U*_{0}, *U*_{1}} and *Y* = min {*U*_{0}, *U*_{2}}.

(ii) (*X*, *Y*) ∼ BIEF(*β*_{0}, *β*_{1}, *β*_{2}; *c*) if and only if there exist independent IEF random variables *U*_{i}, *i* = 0, 1, 2, such that *X* = max {*U*_{0}, *U*_{1}} and *Y* = max {*U*_{0}, *U*_{2}}.

**Theorem 6** The conditional distribution of *X* given *Y* = *y*, is given by

$$ {f}_{X\mid Y}\left(x|y\right)=\left\{\begin{array}{c}\frac{\theta_2\left({\theta}_0+{\theta}_1\right)}{\theta_0+{\theta}_2}\ {g}_1^{\prime}\left(x;c\right)\ {\mathrm{e}}^{-\left({\theta}_0+{\theta}_1\right){g}_1\left(x;c\right)+{\theta}_0{g}_1\left(y;c\right)},\mathrm{if}\ x>y\\ {}{\theta}_1{g}_1^{\prime}\left(x;c\right)\ {\mathrm{e}}^{-{\theta}_1{g}_1\left(x;c\right)},\mathrm{if}\ x<y\\ {}\frac{\theta_0}{\theta_0+{\theta}_2}\ {\mathrm{e}}^{-{\theta}_1{g}_1\left(x;c\right)},\mathrm{if}\ x=y\end{array}\right. $$

(9)

for the BEF, while for the BIEF is given by

$$ {f}_{X\mid Y}\left(x|y\right)=\left\{\begin{array}{c}-{\beta}_1{g}_2^{\prime}\left(x;c\right){e}^{-{\beta}_1{g}_2\left(x;c\right)},\mathrm{if}\ x>y\\ {}\frac{-{\beta}_2\left({\beta}_0+{\beta}_1\right)}{\ \left({\beta}_0+{\beta}_2\right)}{g}_2^{\prime}\left(x;c\right){e}^{-\left({\beta}_0+{\beta}_1\right){g}_2\left(x;c\right)+{\beta}_0{g}_2\left(y;c\right)},\mathrm{if}\ x<y\\ {}\frac{\beta_0}{\beta_0+{\beta}_2}\ {\mathrm{e}}^{-{\beta}_1{g}_2\left(x;c\right)},\mathrm{if}\ x=y\end{array}\right. $$

(10)

**Proof** The proof is trivial so it is omitted.

**Theorem 7** If (*X*, *Y*) is a bivariate vector of continuous random variables, then

- (i)
min{*X*, *Y*} ∼ EF(*θ*; *c*), if (*X*, *Y*) ∼ BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*),

- (ii)
$$ \max \left\{X,Y\right\}\sim \mathrm{IEF}\left(\beta; c\right),\mathrm{if}\ \left(X,Y\right)\sim \mathrm{BIEF}\left({\beta}_0,{\beta}_1,{\beta}_2;c\right). $$

**Proof** (i) if (*X*, *Y*) ∼ BEF(*θ*_{0}, *θ*_{1}, *θ*_{2}; *c*), then using (4), we have

$$ P\left(\min \left\{X,Y\right\}>t\right)=P\left(X>t,Y>t\right)={e}^{-{\theta}_1{g}_1\left(t;c\right)-{\theta}_2{g}_1\left(t;c\right)-{\theta}_0{g}_1\left(t;c\right)}={e}^{-\theta {g}_1\left(t;c\right)}. $$

Similarly by using (5) for the BIEF, we can show that max{*X*, *Y*} ∼ IEF(*β*; *c*). ∎