On the ratio of Pearson type VII and Bessel random variables

For given random variablesX and Y , the distribution of the ratioX/Y is of interest in biological and physical sciences, econometrics, and ranking and selection. Examples include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, and inventory ratios in economics. Another important example is the stress-strength model in the context of reliability. It describes the life of a component which has a random strength Y and is subjected to random stress X . The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever Y > X . Thus, Pr(X < Y) is a measure of a component reliability. It has many applications especially in engineering concepts such as structures, deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft structures and aging of concrete pressure vessels. The distribution of X/Y has been studied by several authors especially when X and Y are independent random variables and come from the same family. For instance, see Marsaglia [14] and Korhonen and Narula [9] for normal family, Press [18] for Student’s t family, Basu and Lochner [1] forWeibull family, Shcolnick [23] for stable family, Hawkins and Han [5] for non-central chi-squared family, Provost [19] for gamma family, and Pham-Gia [17] for beta family. However, there is relatively little work of this kind when X and Y belong to different families. In the applications mentioned above, it is indeed quite possible that X and Y could arise from different but similar distributions. Pearson type VII distributions (which contain Student’s t distributions as particular cases) are becoming of increasing importance in classical as well as in Bayesian statistical