PosRatioDist (version 1.2.1)

dBiMG_expPR: BiMG_expPR

Description

probability density function of quotient of Morgenstern type bivariate exponential random variables conditioned to the positive quadrant.For more detailed information please read the first reference paper.

Usage

dBiMG_expPR(x, a, b, alpha)

Arguments

x

vector of positive quantiles.

a

parameter for Morgenstern type bivariate exponential distribution

b

parameter for Morgenstern type bivariate exponential distribution

alpha

parameter for Morgenstern type bivariate exponential distribution

Value

dBiMG_expPR gives the probability density function for quotient of Morgenstern type bivariate exponential random variables conditioned to the positive quadrant

Invalid arguments will return an error message.

Details

Probability density function $$f_R (r \mid X > 0, Y > 0) = \frac {(1 + \alpha) \exp (a + b)}{\Pr (X > 0, Y > 0) (1 + r)^2} - \frac {2 \alpha \exp (a + 2 b)}{\Pr (X > 0, Y > 0) (2 + r)^2} - \frac {2 \alpha \exp (2 a + b)}{\Pr (X > 0, Y > 0) (1 + 2 r)^2} + \frac {\alpha \exp (2 a + 2 b)}{\Pr (X > 0, Y > 0) (1 + r)^2}$$

For \(r > 0\),\(-1 \leq \alpha \leq 1, a > -\infty, b > -\infty \) These correlated exponential random variables can also be used to model the stress and strength components of a system, hence the quotient distribution can be used to estimate the probability of failure of the system

References

Yuancheng Si and Saralees Nadarajah and Xiaodong Song, (2020). On the distribution of quotient of random variables conditioned to the positive quadrant. Communications in Statistics - Theory and Methods, 49, pp2514-2528.

Balakrishnan, N. and Lai, C. -D. (2009).Continuous Bivariate Distributions.Springer Verlag, New York.

Balakrishna, N. and Shiji, K. (2014).On a class of bivariate exponential distributions.Statistics and Probability Letters, 85, pp153-160.

Examples

# NOT RUN {
x <- seq(0.1,5,0.1)
y <- dBiMG_expPR(x, 3, 2, 0.5)
plot(x,y,type = 'l')


# }