## Some probability density functions and their characteristic functions

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**27**(1973), 495-504 Request permission

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

## References

- B. C. Bhattacharyya,
*The use of McKay’s Bessel function curves for graduating frequency distributions*, Sankhyā**6**(1942), 175–182. MR**9272**
R. C. Bose, "On the exact distribution and moment-coefficients of the ${D^2}$-statistic," - K. V. Krishna Sastry,
*On a Bessel function of the second kind and Wilks’ $Z$-distribution*, Proc. Indian Acad. Sci., Sect. A.**28**(1948), 532–536. MR**0027995** - R. G. Laha,
*On some properties of the Bessel function distributions*, Bull. Calcutta Math. Soc.**46**(1954), 59–72. MR**63617** - R. D. Lord,
*The use of the Hankel transform in statistics. I. General theory and examples*, Biometrika**41**(1954), 44–55. MR**61791**, DOI 10.2307/2333004 - R. D. Lord,
*The use of the Hankel transform in statistics. II. Methods of computation*, Biometrika**41**(1954), 344–350. MR**65090**, DOI 10.2307/2332715 - R. D. Lord,
*The distribution of distance in a hypersphere*, Ann. Math. Statistics**25**(1954), 794–798. MR**65048**, DOI 10.1214/aoms/1177728669 - Yudell L. Luke,
*Integrals of Bessel functions*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR**0141801**
Yudell L. Luke, - Frank McNolty,
*A contour-integral derivation of the non-central chi-square distribution*, Ann. Math. Statist.**33**(1962), 796–800. MR**137195**, DOI 10.1214/aoms/1177704598 - Frank McNolty,
*Applications of Bessel function distributions*, Sankhyā Ser. B**29**(1967), 235–248. MR**0228088**
Frank McNolty, "A note on radial distributions," - Frank McNolty,
*Quadratic form distributions associated with special functions*, Sankhyā Ser. B**34**(1972), 21–26. MR**0326923** - Frank McNolty and Jack Tomsky,
*Some properties of special-function, bivariate distributions*, Sankhyā Ser. B**34**(1972), 251–264. MR**0336873**
Frank McNolty, "Random vectors with non-uniform phase distributions." (Submitted.)
- Frank McNolty,
*Reliability density functions when the failure rate is randomly distributed*, Sankhyā Ser. A**26**(1964), 287–292. MR**187266**
Frank McNolty, R. Clow & E. Hansen, "Some matched filter configurations for infrared systems," - Raj Des,
*On a generalised Bessel function population*, Gaṇita**3**(1953), 111—115. MR**53444** - Norman L. Johnson and Samuel Kotz,
*Distributions in statistics. Continuous univariate distributions. 1.*, Houghton Mifflin Co., Boston, Mass., 1970. MR**0270475** - Norman L. Johnson and Samuel Kotz,
*Distributions in statistics. Continuous univariate distributions. 2.*, Houghton Mifflin Co., Boston, Mass., 1970. MR**0270476**

*Sankhyā*, v. 2, 1936, pp. 143-154. S. S. Bose, "On a Bessel function population,"

*Sankhyā*, v. 3, 1938, pp. 253-261. A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,

*Tables of Integral Transforms*, Vol. 1, McGraw-Hill, New York and London, 1954. MR

**15**, 868. J. O. Irwin, "On the frequency distribution of the means of samples,"

*Biometrika*, v. 19, 1927, pp. 225-239.

*The Spectral Functions and Their Approximations*, Vol. 1, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR

**39**#3039. Yudell L. Luke,

*The Special Functions and Their Approximations*. Vol. 2, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR

**40**#2909. A. T. McKay, "A Bessel function distribution,"

*Biometrika*, v. 24, 1931, pp. 39-44.

*Operations Res.*, v. 16, 1968a, pp. 211-216. Frank McNolty, "Expected coverage for targets of nonuniform density,"

*Operations Res.*, v. 16, 1968b, pp. 1027-1040.

*IEEE Trans. Aerospace and Electronic Systems*, v. AES-8, 1972, pp. 428-438. Frank McNolty, R. Clow & E. Hansen, "Some properties of the output of integrator in an infrared system,"

*IEEE Trans. Aerospace and Electronic Systems*, v. AES-8, 1972, pp. 552-558. Karl Pearson, "Further applications in statistics of the ${T_m}(X)$ Bessel function,"

*Biometrika*, v. 24, 1932, pp. 293-350.

## Additional Information

- © Copyright 1973 American Mathematical Society
- 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