Some probability density functions and their characteristic functions

Abstract:

This paper presents, without derivation, several generalized density functions together with their characteristic functions. The densities are expressed variously in terms of special functions such as: ${I_v}(x)$, the modified Bessel function of the first kind of order v; ${K_v}(x)$, the modified Bessel function of the second kind of order v; $_1{F_1}(a;b;x)$, the confluent hypergeometric function; $_2{F_1}(a,b;c;x)$, the hypergeometric function; ${W_{a,b}}(x)$, Whittaker’s function; ${\Phi _3}(\beta ;\gamma ;bx,cx)$, a generalized hypergeometric function (type I); \[ {\Phi _2}(b,c,d;\gamma ;\lambda x,\tau x,\beta x),\] a generalized hypergeometric function (type II); and $\phi _\lambda ^\mu (b{v^\mu })$, a generalized Bessel type function. The first five cases are summarized from the work of Laha [7], Pearson [25] and Raj [26] while Cases 13 through 19 have not previously appeared in the literature of statistics or Fourier transforms. In what follows, the usual notation $f(x)$, for a density function, and $\varphi (t)$, for a characteristic function, will be used with all parameters considered as real quantities: \[ \varphi (t) = \int _{ - \infty }^\infty {\exp (itx)f(x)\;dx.} \]