probability density function of quotient of Bivariate Lomax random variables conditioned to the positive quadrant.For more detailed information please read the first reference paper.
dBilomaxPR(x, a, b, c, alpha, beta, theta)single positive scalar for quotient
parameter for Bivariate lomax distribution
parameter for Bivariate lomax distribution
parameter for Bivariate lomax distribution
parameter for Bivariate lomax distribution
parameter for Bivariate lomax distribution
parameter for Bivariate lomax distribution
dBilomaxPR gives the probability density function for bivariate lomax random variables conditioned to the positive quadrant.
Invalid arguments will return an error message.
Probability density function $$f_R (r \mid X > 0, Y > 0) = \frac {c^2 \theta^2 r}{\Pr (X > 0, Y > 0)} J_3 \left( \theta r, \beta - \theta a + \left( \alpha - \theta b \right) r, 1 - \alpha a - \beta b + \theta a b, c + 2 \right) +\frac {c^2 \theta \left[ (\alpha - \theta b) r + \beta - \theta a \right]} {\Pr (X > 0, Y > 0)} J_2 \left( \theta r, \beta - \theta a + \left( \alpha - \theta b \right) r, 1 - \alpha a - \beta b + \theta a b, c + 2 \right) +\frac {c \left[ c (\alpha - \theta b) (\beta - \theta a) + \alpha \beta - \theta \right]}{\Pr (X > 0, Y > 0)}J_1 \left( \theta r, \beta - \theta a + \left( \alpha - \theta b \right) r,1 - \alpha a - \beta b + \theta a b, c + 2 \right)$$
For \(r > 0\),\(\alpha > 0\), \(\beta > 0\), \(\theta > 0\), \(0 \leq \theta \leq (c + 1) \alpha \beta\) where \(J_1,J_2,J_3\) are given by first reference paper section (2.5)
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.