On the Product and Ratio of Laplace and Bessel Random Variables

For given random variables X and Y , the distributions of the product |XY | and the ratio |X/Y | are of interest in many areas of the sciences. In traditional portfolio selection models, certain cases involve the product of random variables. The best examples of this are in the case of investment in a number of different overseas markets. In portfolio diversification models (see, e.g., Grubel [7]), not only are prices of shares in local markets uncertain but also the exchange rates are uncertain so that the value of the portfolio in domestic currency is related to a product of random variables. Similarly in models of diversified production by multinationals (see, e.g., Rugman [23]), there is local production uncertainty and exchange rate uncertainty so that profits in home currency are again related to a product of random variables. An entirely different example is drawn from the econometric literature. In making a forecast from an estimated equation, Feldstein [5] pointed out that both the parameter and the value of the exogenous variable in the forecast period could be considered as random variables. Hence, the forecast was proportional to a product of random variables. An important example of ratios of random variables 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 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 the aging of concrete pressure vessels.