Exact Kolmogorov and total variation distances between some familiar discrete distributions

Open Access
Research Article


We give exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson, binomial, and negative binomial distributions with different parameters. In the Poisson case, such expressions are related with the Lambert Open image in new window function.


Total Variation Binomial Distribution Variation Distance Negative Binomial Distribution Discrete Distribution 


  1. 1.
    Adell JA, Lekuona A: Sharp estimates in signed Poisson approximation of Poisson mixtures. Bernoulli 2005,11(1):47–65. 10.3150/bj/1110228242MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Barry DA, Parlange J-Y, Li L, Prommer H, Cunningham CJ, Stagnitti F: Analytical approximations for real values of the Lambert-function. Mathematics and Computers in Simulation 2000,53(1–2):95–103. 10.1016/S0378-4754(00)00172-5MathSciNetCrossRefGoogle Scholar
  3. 3.
    Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE: On the Lambertfunction. Advances in Computational Mathematics 1996,5(4):329–359.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Johnson NL, Kotz S, Kemp AW: Univariate Discrete Distributions, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. 2nd edition. John Wiley & Sons, New York; 1992:xxii+565.Google Scholar
  5. 5.
    Kennedy JE, Quine MP: The total variation distance between the binomial and Poisson distributions. The Annals of Probability 1989,17(1):396–400. 10.1214/aop/1176991519MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Roos B: Binomial approximation to the Poisson binomial distribution: the Krawtchouk expansion. Theory of Probability and Its Applications 2001,45(2):258–272. 10.1137/S0040585X9797821XMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Roos B: Improvements in the Poisson approximation of mixed Poisson distributions. Journal of Statistical Planning and Inference 2003,113(2):467–483. 10.1016/S0378-3758(02)00095-2MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ruzankin PS: On the rate of Poisson process approximation to a Bernoulli process. Journal of Applied Probability 2004,41(1):271–276. 10.1239/jap/1077134684MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© J. A. Adell and P. Jodrá 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Departamento de Métodos EstadísticosUniversidad de ZaragozaZaragozaSpain

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