Some Probability Density Functions and Their Characteristic Functions

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: \[ \varphi (t) = \int _{ - \infty }^\infty {\exp (itx)f(x)\;dx.} \]