• Published : 2005.07.01


The distribution of products of random variables is of interest in many areas of the sciences, engineering and medicine. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the distribution of the product | XY | is derived when X and Y are Student's t and Bessel function random variables distributed independently of each other.


t distribution;Bessel function distribution;products of random variables


  1. M. S. Abu-Salih, Distributions of the product and the quotient of power-function random variables, Arab J. Math. Sci. 4 (1983), 77-90
  2. J. W. Atkinson, An Introduction to Motivation, Princeton: Van Nostrand, 1964
  3. R. P. Bhargava and C. G. Khatri, The distribution of product of independent beta random variables with application to multivariate analysis, Ann. Inst. Statist. Math. 33 (1981), 287-296
  4. H. K. Cigizoglu and M. Bayazit, A generalized seasonal model for flow duration curve, Hydrological Processes, 14 (2000), 1053-1067<1053::AID-HYP996>3.0.CO;2-B
  5. R. L. Decker, A study of three specific problems in the measurement and interpretation of employee attributes, Psychological Monographs, 69 (1955), no 401
  6. R. B. Ewen, Weighting components of job satisfaction, Journal of Applied Psychology, 51 (1967), 68-73
  7. M. S. Feldstein, The error of forecast in econometric models when the forecastperiod exogenous variables are stochastic, Econometrica, 39 (1971)
  8. J. H. Goldthorpe, et al, The Ajfiuent Worker: Industrial Attitudes and Behavior, Cambridge: Cambridge University Press, 1968
  9. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products(sixth edition), San Diego: Academic Press, 2000
  10. H. G. Grubel, Internationally diversified portfolios: welfare gains capital flows, American Economic Review, 58 (1968)
  11. A. L. Kalleberg, Work values, job rewards, and job satisfaction: A theory of the equality of work experience, Ph.D. Dissertation, University of Wisconsin, 1975
  12. A. L. Kalleberg, Work values and job rewards: A theory of job satisfaction, American Sociological Review, 42 (1977), 124-143
  13. S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering and Finance, Boston: Birkhauser, 2001
  14. S. Kotz and S. Nadarajah, Multivariate t Distributions and Their Applications, New York: Cambridge University Press, 2004
  15. S. Nadarajah and S. Kotz, Skewed distributions generated by the normal kernel, Statist. Probab. Lett. 65 (2003), 269-277
  16. E. Palmore and P. E. Hammond, Interacting factor in juvenile delinquency, American Sociological Review, 29 (1964), 848-854
  17. H. Podolski, The distribution of a product of n independent random variables with generalized gamma distribution, Demonstratio Math. 4 (1972), 119-123
  18. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, 1, 2 and 3 (1986), Amsterdam: Gordon and Breach Science Publishers
  19. P. N. Rathie and H. G. Rohrer, The exact distribution of products of independent random variables, Metron, 45 (1987), 235-245
  20. M. Rokeach and P. Kliejunas, Behavior as a function of attitude-toward-object and attitude-toward-situation, Journal of Personality and Social Psychology, 22 (1972), 194-201
  21. A. M. Rugman, International Diversification and the Multinational Enterprise, Lexington, 1979
  22. H. Sakamoto, On the distributions of the product and the quotient of the independent and uniformly distributed random variables, Tohoku Math. J. 49 (1943), 243-260
  23. R. H. Schaffer, Job satisfaction as related to need satisfaction at work, Psychological Monographs 67 (1953), no. 364
  24. M. D. Springer and W. E. Thompson, The distribution of products of beta, gamma and Gaussian random variables, SIAM J. Appl. Math. 18 (1970), 721-737
  25. B. M. Steece, On the exact distribution for the product of two independent betadistributed random variables, Metron, 34 (1976), 187-190
  26. A. Stuart, Gamma-distributed products of independent random variables, Biometrika, 49 (1962), 564-565
  27. J. Tang and A. K. Gupta, On the distribution of the product of independent beta random variables, Statist. Probab. Lett. 2 (1984), 165-168
  28. L. Thabane and S. Drekic, Properties of the non-central t-Bessel distribution and its applications, J. Stat. Theory Appl. 1 (2002), 163-175
  29. C. M. Wallgren, The distribution of the product of two correlated t variates, J. Amer. Statist. Assoc. 75 (1980), 996-1000
  30. L. K. Waters, The utility of importance weights in predicting overall job satisfaction and dissatisfaction, Educational and Psychological Measurement, 29 (1969), 519-522
  31. H. L. Harter, On the distribution of Wald's classification statistic, Ann. Statist. 22 (1951), 58-67
  32. M. Ladekarl, V. Jensen, and B. Nielsen, Total number of cancer cell nuclei and mitoses in breast tumors estimated by the optical dis ector, Analytical and Quantitative Cytology and Histology, 19 (1997), 329-337
  33. H. J. Malik and R. Trudel, Probability density function of the product and quotient of two correlated exponential random variables, Canad. Math. Bull. 29 (1986), 413-418