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Table 1 Summary of some special cases of the BEF(θ012;c) or BIEF(β012;c)

From: Bivariate general exponential models with stress-strength reliability application

Distribution

BEF(θ0, θ1, θ2; c)

BIEF(β0, β1, β2; c)

R

θi

g1(x; c)

βi

g2(x; c)

M-O bivariate exponential

θi

x

_

_

\( \frac{\theta_2}{\theta_0+{\theta}_1+{\theta}_2} \), see [7]

Bivariate Rayleigh

λi

x2

_

_

\( \frac{\lambda_2}{\lambda_0+{\lambda}_1+{\lambda}_2} \), see [9]

M-O bivariate Weibull

\( \frac{1}{{\theta_i}^c} \)

xc

_

_

\( \frac{\theta_0^c{\theta}_1^c}{\theta_0^c+{\theta}_1^c+{\theta}_2^c} \)

M-O Pareto

ai

\( \ln \left(\frac{x}{c}\right) \)

_

_

\( \frac{a_2}{a_0+{a}_1+{a}_2} \)

Bivariate inverse Weibull

_

_

\( \frac{1}{{\theta_i}^c} \)

xc

\( \frac{\theta_0^c{\theta}_2^c}{\theta_0^c+{\theta}_1^c+{\theta}_2^c} \)

Bivariate Burr type III

_

_

bi

\( \ln \left(\frac{1+{x}^c}{x^c}\right) \)

\( \frac{b_1}{b_0+{b}_1+{b}_2} \)