ON THE PRODUCT AND RATIO OF BESSEL RANDOM VARIABLES

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product | XY | and the ratio | X/Y | are derived when X and Y are independent Bessel function random variables. An application of the results is provided by tabulating the associated percentage points.


Introduction
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 these are in the case of investment in a number of different overseas markets.In portfolio diversification models (see, e.g., Grubel [6]), 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 [21]), there are 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 [4] 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.
The distributions of |XY | and |X/Y | have been studied by several authors especially when X and Y are independent random variables and come from the same family.With respect to products of random variables, see Sakamoto [22] for uniform family, Harter [7] and Wallgren [28] for Student's t family, Springer and Thompson [24] for normal family, Stuart [26] and Podolski [14] for gamma family, Steece [25], Bhargava and Khatri [3], and Tang and Gupta [27] for beta family, Abu-Salih [1] for power function family, and Malik and Trudel [11] for exponential family (see also Rathie and Rohrer [20] for a comprehensive review of known results).With respect to ratios of random variables, see Marsaglia [12] and Korhonen and Narula [9] for normal family, Press [15] for Student's t family, Basu and Lochner [2] for Weibull family, Shcolnick [23] for stable family, Hawkins and Han [8] for noncentral chi-square family, Provost [16] for gamma family, and Pham-Gia [13] for beta family.
In this paper, we study the exact distributions of and n > 1, where is the modified Bessel function of the third kind.Tabulations of the associated percentage points are also provided.Bessel function distributions have found applications in a variety of areas that range from image and speech recognition and ocean engineering to finance.They are rapidly becoming distributions of first choice whenever "something" heavier than Gaussian tails is observed in the data.Some examples are as follows (see Kotz et al. [10] for further applications).

Product
Theorem 2.1 derives explicit expressions for the distribution of |XY | in terms of the 0 F 3 and 1 F 4 hypergeometric functions.
Theorem 2.1.Suppose that X and Y are distributed according to (1.1) and (1.2), respectively.The pdf and the cdf of Z = |XY | can be expressed as S. Nadarajah and A. K. Gupta 2981 where and C denotes Euler's constant.
Proof.The pdf of |XY | can be expressed as where I(m,n) denotes the integral The result in (2.1) follows by direct application of Lemma 1.1 to calculate I(m,n).The cdf of Z can be expressed as The result in (2.2) follows by applying Lemma 1.4 to calculate the three integrals in (2.6). where Proof.The pdf of Z = |X/Y | can be expressed as where I(m,n) denotes the integral The result in (3.1) follows by direct application of Lemma 1.2 to calculate I(m,n).The cdf of Z can be expressed as Using special properties of the 2 F 1 hypergeometric function, one can derive other equivalent forms and elementary forms for the pdf of Z = |X/Y |.This is illustrated in the corollaries below.
Corollary 3.2.The pdf given by (3.1) can be expressed in the equivalent forms (3.9) the equations  We hope these numbers will be of use to the practitioners mentioned in Section 1. Similar tabulations could be easily derived for other values of p, m, n, b, and β by using the hypergeom (•) function in Maple.

)
In communication theory, X and Y could represent the random noises corresponding to two different signals.(2) In ocean engineering, X and Y could represent distributions of navigation errors.(3) In finance, X and Y could represent distributions of log-returns of two different commodities.(4) In image and speech recognition, X and Y could represent "input" distributions.In each of the examples above, it will be of interest to study the distribution of the ratio |X/Y |.For example, in communication theory, |X/Y | could represent the relative strength of the two different signals.In ocean engineering, |X/Y | could represent the relative safety of navigation.In finance, |X/Y | could represent the relative popularity of the two different commodities.The distribution of the product |XY | is considered here for completeness.S. Nadarajah and A. K. Gupta 2979

Figure 2 .Theorem 3 . 1 .
Figure 2.1 illustrates possible shapes of the pdf (2.1) for selected values of m and n.The four curves in each plot correspond to selected values of n.The effect of the parameters is evident.

Figure 4 .Figure 4 . 1 .
Figure 4.1 illustrates possible shapes of the pdf (3.1) for selected values of m and n.The four curves in each plot correspond to selected values of n.The effect of the parameters is evident.In this section, we provide tabulations of percentage points associated with the derived distributions of |XY | and |X/Y |.These values are obtained by numerically solving |XY | and |X/Y | when X and Y are independent Bessel function random variables with pdfs
Evidently, this involves computation of the generalized hypergeometric function and routines for this are widely available.We used the function hypergeom (•) in the algebraic manipulation package Maple.Tables 4.1 and 4.2 provide the numerical values of z p for b = 1, β = 1, m = 2,3,...,9, and n = m,m + 1,...,9.