Table 1 Summary of some special cases of the BEF(θ0,θ1,θ2;c) or BIEF(β0,β1,β2;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 | _ | _ | \( \frac{1}{{\theta_i}^c} \) | x−c | \( \frac{\theta_0^c{\theta}_2^c}{\theta_0^c+{\theta}_1^c+{\theta}_2^c} \) |
Bivariate Burr type III | _ | _ | bi | \( \ln \left(\frac{1+{x}^c}{x^c}\right) \) | \( \frac{b_1}{b_0+{b}_1+{b}_2} \) |