On the Ratios of Pathway Random Variables

Abstract

A study of the densities of ratios of independently distributed random variables following the pathway model, Mathai (2005), is carried out. The density functions of these random variables are obtained in terms of H-function. The particular cases of the integral forms are shown to be associated with Tsallis statistics and Beck–Cohen superstatistics. Many other special functions coming under the general density are also included. We plotted the density function of the ratio of these random variables for the different values of the pathway parameters. Real-life application of the results in communication theory related to the signal–noise ratio is illustrated.

Keywords:

• H-function
• Mellin transform
• Pathway model
• Ratios of random variables

Mathematics Subject Classification:

• 62E15
• 60E05
• 33C60
• 44A15

Appendix

The H-function is defined as a Mellin–Barnes type integral as follows: (A.1) where , z ≠ 0 and (A.2)

An empty product is interpreted as unity; m, n, p, q ∈ N₀ with 0 ⩽ n ⩽ p, 1 ⩽ m ⩽ q,α_i, β_j ∈ R +, a_i, b_j ∈ C, i = 1, ..., p; j = 1, ..., q such that α_i(b_j + k) ≠ β_j(a_i − λ − 1), k, λ ∈ N₀; i = 1, 2, ..., n; j = 1, 2, ..., m, where we employ the usual notations: N₀ = 0, 1, 2, ...; R + =(0, ∞), and C being the complex number field. Here L is a suitable contour separating the poles of Γ(b_j + β_js), j = 1, 2, ..., m from the poles of Γ(1 − a_i − α_is), i = 1, 2, ..., n. The contour L could be L = L_{− ∞} or L = L_{+ ∞} or L = L_iγ∞, where L_{− ∞} is a loop starting at − ∞ encircling all the poles of Γ(b_j + β_js), j = 1, 2, ..., m and ending at − ∞. L_{+ ∞} is a loop starting at + ∞ encircling all the poles of Γ(1 − a_i − α_is), i = 1, 2, ..., n and ending at + ∞. L_iγ∞ is the infinite semicircle starting at γ − i∞ and going to γ + i∞ where γ ∈ R = ( − ∞, +∞). When α₁ = α₂ = ... = α_p = 1 = β₁ = β₂ = ... = β_q, H-function reduces to G-function; see Brychkov (2008), Mathai et al. (2010), Srivastava et al. (1982).

A general hypergeometric series with p upper parameters and q lower parameters is defined as follows: (A.3) Here, the shifted factorial, or the Pochhammer symbol, denoted by (a)_n is defined by (A.4)