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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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: , the modified Bessel function of the first kind of order *v*; , the modified Bessel function of the second kind of order *v*; , the confluent hypergeometric function; , the hypergeometric function; , Whittaker's function; , a generalized hypergeometric function (type I);

**[1]**B. C. Bhattacharyya, "The use of McKay's Bessel function curves for graduating frequency distributions,"*Sankhyā*, v. 6, 1942, pp. 175-182. MR**5**, 126. MR**0009272 (5:126e)****[2]**R. C. Bose, "On the exact distribution and moment-coefficients of the -statistic,"*Sankhyā*, v. 2, 1936, pp. 143-154.**[3]**S. S. Bose, "On a Bessel function population,"*Sankhyā*, v. 3, 1938, pp. 253-261.**[4]**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.**[5]**J. O. Irwin, "On the frequency distribution of the means of samples,"*Biometrika*, v. 19, 1927, pp. 225-239.**[6]**K. V. Krishna Sastry, "On a Bessel function of the second kind and Wilks Z-distribution,"*Indian Acad. Sci. Ser. A*, v. 28, 1948, pp. 532-536. MR**10**, 387. MR**0027995 (10:387c)****[7]**Radha G. Laha, "On some properties of the Bessel function distributions,"*Bull. Calcutta Math Soc.*, v. 46, 1954, pp. 59-72. MR**16**, 152. MR**0063617 (16:152f)****[8]**R. D. Lord, "The use of Hankel transforms in statistics. I,"*Biometrika*, v. 41, 1954a, pp. 44-55. MR**15**, 885. MR**0061791 (15:885d)****[9]**R. D. Lord, "The use of the Hankel transform in statistics. II,"*Biometrika*, v. 41, 1954b, pp. 344-350. MR**16**, 382. MR**0065090 (16:382c)****[10]**R. D. Lord, "The distribution of distance in a hypersphere,"*Ann. Math. Statist.*, v. 25, 1954c, pp. 794-798. MR**16**, 377. MR**0065048 (16:377d)****[11]**Yudell L. Luke,*Integrals of Bessel Functions*, McGraw-Hill, New York and London, 1962. MR**25**#5198. MR**0141801 (25:5198)****[12]**Yudell L. Luke,*The Spectral Functions and Their Approximations*, Vol. 1, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR**39**#3039.**[13]**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.**[14]**A. T. McKay, "A Bessel function distribution,"*Biometrika*, v. 24, 1931, pp. 39-44.**[15]**Frank McNolty, "A contour-integral derivation of the non-central chi-square distribution,"*Ann. Math. Statist.*, v. 33, 1962, pp. 796-800. MR**25**#651. MR**0137195 (25:651)****[16]**Frank McNolty, "Applications of Bessel function distributions,"*Sankhyā Ser. B*, v. 29, 1967, pp. 235-248. MR**37**#3672. MR**0228088 (37:3672)****[17]**Frank McNolty, "A note on radial distributions,"*Operations Res.*, v. 16, 1968a, pp. 211-216.**[18]**Frank McNolty, "Expected coverage for targets of nonuniform density,"*Operations Res.*, v. 16, 1968b, pp. 1027-1040.**[19]**Frank McNolty, "Quadratic form distribution associated with special functions,"*Sankhyā Ser. B*, v. 34, 1972, pp. 21-26. MR**0326923 (48:5265)****[20]**Frank McNolty & Jack Tomsky, "Some properties of special-function, bivariate distributions,"*Sankhyā Ser. B*, v. 34, 1972, pp. 251-264. MR**0336873 (49:1646)****[21]**Frank McNolty, "Random vectors with non-uniform phase distributions." (Submitted.)**[22]**Frank McNolty, "Reliability density functions when the failure rate is randomly distributed,"*Sankhyā Ser. A*, v. 26, 1964, pp. 287-292. MR**32**#4719. MR**0187266 (32:4719)****[23]**Frank McNolty, R. Clow & E. Hansen, "Some matched filter configurations for infrared systems,"*IEEE Trans. Aerospace and Electronic Systems*, v. AES-8, 1972, pp. 428-438.**[24]**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.**[25]**Karl Pearson, "Further applications in statistics of the Bessel function,"*Biometrika*, v. 24, 1932, pp. 293-350.**[26]**Des Raj, "On a generalized Bessel function population,"*Ganita*, v. 3, 1953, pp. 111-115. MR**14**, 775. MR**0053444 (14:775f)****[27]**Norman L. Johnson & Samuel Kotz,*Distributions in Statistics. Continuous Univariate Distributions*. 1, Houghton Mifflin, Boston, Mass., 1970. MR**42**#5363. MR**0270475 (42:5363)****[28]**Norman L. Johnson & Samuel Kotz,*Distributions in Statistics. Continuous Univariate Distributions*. 2, Houghton Mifflin, Boston, Mass., 1970. MR**42**#5364. MR**0270476 (42:5364)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65C99,
60E05

Retrieve articles in all journals with MSC: 65C99, 60E05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0329193-3

Article copyright:
© Copyright 1973
American Mathematical Society