Bivariate distributions of some ratios of independent noncentral chi-square random variables

On the Product and Ratio of Gamma and Beta Random Variables

“…With respect to XY , see Sakamoto (1943) for uniform family, Harter (1951) and Wallgren (1980) for Student's t family, Springer and Thompson (1970) for normal family, Stuart (1962) and Podolski (1972) for gamma family, Steece (1976), Bhargava and Khatri (1981) and Tang and Gupta (1984) for beta family, AbuSalih (1983) for power function family, and Malik and Trudel (1986) for exponential family (see also Rathie and Rohrer (1987) for a comprehensive review of known results). With respect to X/Y , see Marsaglia (1965) and Korhonen and Narula (1989) for normal family, Press (1969) for Student's t family, Basu and Lochner (1971) for Weibull family, Shcolnick (1985) for stable family, Hawkins and Han (1986) for non-central chi-squared family, Provost (1989) for gamma family, and Pham-Gia (2000) for beta family. There is relatively little work of this kind when X and Y belong to different families.…”

On the ratios for extreme value distributions with application to rainfall modeling

“…For instance, see Marsaglia (1965) and Korhonen and Narula (1989) for the normal family, Press (1969) for the Student's-t family, Basu and Lochner (1971) for the Weibull family, Shcolnick (1985) for the stable family, Hawkins and Han (1986) for the non-central chi-squared family, Provost (1989) for the gamma family, and Pham-Gia (2000) for the beta family.…”

Ratio of generalized random variables with application

“…and Lochner (1971) for Weibull family, Shcolnick (1985) for stable family, Hawkins and Han (1986) for noncentral chi-squared family, Provost (1989) for gamma family, and Pham-Gia (2000) for beta family. This note represents the first study of the distribution of X/(X+Y) when X and Y are independent generalized Pareto random variables.…”