Sum, product and ratio of Pareto and gamma variables

References

• Fisher , R. A. 1935 . The fiducial argument in statistical inference . Ann. Eugenics , 6 : 391 – 398 .
   Google Scholar
• Chapman , D. G. 1950 . Some two-sample tests . Ann. Math. Statist. , 21 : 601 – 606 .
   Google Scholar
• Christopeit , N. and Helmes , K. 1979 . A convergence theorem for random linear combinations of independent normal random variables . Ann. Statist. , 7 : 795 – 800 .
   Web of Science ®Google Scholar
• Davies , R. B. 1980 . Algorithm AS 155: the distribution of a linear combination of chi-squared random variables . Appl. Statist. , 29 : 323 – 333 .
   Google Scholar
• Farebrother , R. W. 1984 . Algorithm AS 204: the distribution of a positive linear combination of chi-squared random variables . Appl. Statist. , 33 : 332 – 339 .
   Web of Science ®Google Scholar
• Ali , M. M. 1982 . Distribution of linear combinations of exponential variates . Commun. Statist. Theory Meth. , 11 : 1453 – 1463 .
   Web of Science ®Google Scholar
• Moschopoulos , P. G. 1985 . The distribution of the sum of independent gamma random variables . Ann. Inst. Statist. Math. , 37 : 541 – 544 .
   Web of Science ®Google Scholar
• Provost , S. B. 1989 . On sums of independent gamma random variables . Statistics , 20 : 583 – 591 .
   Google Scholar
• Dobson , A. J. , Kulasmaa , K. and Scherer , J. 1991 . Confidence intervals for weighted sums of Poisson parameters . Statist. Med. , 10 : 457 – 462 .
   PubMed Web of Science ®Google Scholar
• Pham–Gia , T. and Turkkan , N. 1993 . Bayesian analysis of the difference of two proportions . Commun. Statist. Theory Meth. , 22 : 1755 – 1771 .
   Web of Science ®Google Scholar
• Pham , T. G. and Turkkan , N. 1994 . Reliability of a standby system with beta-distributed component lives . IEEE Trans. Reliab. , 43 : 71 – 75 .
   Web of Science ®Google Scholar
• Kamgar–Parsi , B. , Kamgar-Parsi , B. and Brosh , M. 1995 . Distribution and moments of weighted sum of uniform random variables with applications in reducing Monte Carlo simulations . J. Statist. Comput. Simul. , 52 : 399 – 414 .
   Web of Science ®Google Scholar
• Albert , J. 2002 . Sums of uniformly distributed variables: a combinatorial approach . College Math. J. , 33 : 201 – 206 .
   Google Scholar
• Hitezenko , P. 1998 . A note on a distribution of weighted sums of iid Rayleigh random variables, Sankhyā, A, 60 . : 171 – 175 .
   Google Scholar
• Hu , C.-Y. and Lin , G. D. 2001 . An inequality for the weighted sums of pairwise iid generalized Rayleigh random variables . J. Statist. Plann. Inference , 92 : 1 – 5 .
   Web of Science ®Google Scholar
• Witkovskay , V. 2001 . Computing the distribution of a linear combination of inverted gamma variables . Kybernetika , 37 : 79 – 90 .
   Google Scholar
• Sakamoto , H. 1943 . On the distributions of the product and the quotient of the independent and uniformly distributed random variables . Tohoku Math. J. , 49 : 243 – 260 .
   Google Scholar
• Harter , H. L. 1951 . On the distribution of Wald's classification statistic . Ann. Math. Statist. , 22 : 58 – 67 .
   Google Scholar
• Wallgren , C. M. 1980 . The distribution of the product of two correlated t variates . J. Amer. Statist. Assoc. , 75 : 996 – 1000 .
   Web of Science ®Google Scholar
• Springer , M. D. and Thompson , W. E. 1970 . The distribution of products of beta, gamma and Gaussian random variables . SIAM J. Appl. Math. , 18 : 721 – 737 .
   Web of Science ®Google Scholar
• Stuart , A. 1962 . Gamma-distributed products of independent random variables . Biometrika , 49 : 564 – 565 .
   Web of Science ®Google Scholar
• Podolski , H. 1972 . The distribution of a product of n independent random variables with generalized gamma distribution . Demonstratio Math. , 4 : 119 – 123 .
   Google Scholar
• Steece , B. M. 1976 . On the exact distribution for the product of two independent beta-distributed random variables . Metron , 34 : 187 – 190 .
   Google Scholar
• Bhargava , R. P. and Khatri , C. G. 1981 . The distribution of product of independent beta random variables with application to multivariate analysis . Ann. Inst. Statist. Math. , 33 : 287 – 296 .
   Web of Science ®Google Scholar
• Tang , J. and Gupta , A. K. 1984 . On the distribution of the product of independent beta random variables . Statist. Probab. Lett. , 2 : 165 – 168 .
   Web of Science ®Google Scholar
• Abu–Salih , M. S. 1983 . Distributions of the product and the quotient of power-function random variables . Arab J. Math. , 4 : 77 – 90 .
   Google Scholar
• Malik , H. I. and Trudel , R. 1986 . Probability density function of the product and quotient of two correlated exponential random variables . Canad. Math. Bull. , 29 : 413 – 418 .
   Web of Science ®Google Scholar
• Rathie , P. N. and Rohrer , H. G. 1987 . The exact distribution of products of independent random variables . Metron , 45 : 235 – 245 .
   Google Scholar
• Marsaglia , G. 1965 . Ratios of normal variables and ratios of sums of uniform variables . J. Amer. Statist. Assoc. , 60 : 193 – 204 .
   Web of Science ®Google Scholar
• Korhonen , P. J. and Narula , S. C. 1989 . The probability distribution of the ratio of the absolute values of two normal variables . J. Statist. Comput. Simul. , 33 : 173 – 182 .
   Google Scholar
• Press , S. J. 1969 . The t ratio distribution . J. Amer. Statist. Assoc. , 64 : 242 – 252 .
   Web of Science ®Google Scholar
• Basu , A. P. and Lochner , R. H. 1971 . On the distribution of the ratio of two random variables having generalized life distributions . Technometrics , 13 : 281 – 287 .
   Web of Science ®Google Scholar
• Shcolnick , S. M. 1985 . “ On the ratio of independent stable random variables, Stability Problems for Stochastic Models (Uzhgorod, 1984) ” . 349 – 354 . Berlin : Springer . Lecture Notes in Mathematics, 1155
   Google Scholar
• Hawkins , D. L. and Han , C.-P. 1986 . Bivariate distributions of some ratios of independent noncentral chi-square random variables . Commun. Statist. Theory Meth. , 15 : 261 – 277 .
   Web of Science ®Google Scholar
• Provost , S. B. 1989 . On the distribution of the ratio of powers of sums of gamma random variables . Pakistan J. Statist. , 5 : 157 – 174 .
   Google Scholar
• Pham–Gia , T. 2000 . Distributions of the ratios of independent beta variables and applications . Commun. Statist. Theory Meth. , 29 : 2693 – 2715 .
   Web of Science ®Google Scholar
• Kim , H. , Hou , J. C. , Hu , C. and Ge , Y. 2007 . QoS provisioning in IEEE 802.11–compliant networks: past, present, and future . Comput. Networks, , 51 ( 8 ) : 1922 – 1941 .
   Google Scholar
• Li , J. , Yuksel , M. and Kalyanaraman , S. 2006 . Explicit rate multicast congestion control . Comput. Networks , 50 : 2614 – 2640 .
   Web of Science ®Google Scholar
• Ma , M. 2006 . Tabu marking scheme to speedup IP traceback . Comput. Networks. , 50 : 3536 – 3549 .
   Google Scholar
• Wu , G. , Chong , E. K.P. and Givan , R. 2006 . Predictive buffer control in delivering remotely stored video using proxy servers . Computer Networks , 50 : 3721 – 3742 .
   Google Scholar
• Nadarajah , S. and Kotz , S. 2008 . The cycle time distribution . Int. J. Production Res. , 46 : 3133 – 3141 .
   Web of Science ®Google Scholar
• Nadarajah , S. 2007 . Economic models based on Pareto and Gamma random variables . Spanish Econom. Rev. , 9 : 1435 – 5477 .
   Google Scholar
• Balakrishnan , N. and Basu , A. P. 1995 . “ The Exponential Distribution: Theory, Methods and Applications ” . Amsterdam : Gordon and Breach Publishers .
   Google Scholar
• Nadarajah , S. 2007 . Comparison of time–to–event data for clinical trials . Monte Carlo Methods Appl. , 13 : 21 – 35 .
   Google Scholar
• Prudnikov , A. P. , Brychkov , Y. A. and Marichev , O. I. 1986 . Integrals and Series , Vol. 1–3 , Amsterdam : Gordon and Breach Science Publishers .
   Google Scholar
• Gradshteyn , I. S. and Ryzhik , I. M. 2000 . Table of Integrals, Series, and Products , sixth , San Diego : Academic Press .
   Google Scholar