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Some probability density functions and their characteristic functions


Author: Frank McNolty
Journal: Math. Comp. 27 (1973), 495-504
MSC: Primary 65C99; Secondary 60E05
DOI: https://doi.org/10.1090/S0025-5718-1973-0329193-3
MathSciNet review: 0329193
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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);

$\displaystyle {\Phi _2}(b,c,d;\gamma ;\lambda x,\tau x,\beta x),$

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:

$\displaystyle \varphi (t) = \int_{ - \infty }^\infty {\exp (itx)f(x)\;dx.} $


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0329193-3
Article copyright: © Copyright 1973 American Mathematical Society

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