Binomial sum variance inequality

The binomial sum variance inequality states that the variance of the sum of binomially distributed random variables will always be less than or equal to the variance of a binomial variable with the same n and p parameters. In probability theory and statistics, the sum of independent binomial random variables is itself a binomial random variable if all the component variables share the same success probability. If success probabilities differ, the probability distribution of the sum is not binomial.[1] The lack of uniformity in success probabilities across independent trials leads to a smaller variance.[2][3][4][5][6] and is a special case of a more general theorem involving the expected value of convex functions.[7] In some statistical applications, the standard binomial variance estimator can be used even if the component probabilities differ, though with a variance estimate that has an upward bias.

  1. ^ Butler, Ken; Stephens, Michael (1993). "The distribution of a sum of binomial random variables" (PDF). Technical Report No. 467. Department of Statistics, Stanford University. Archived (PDF) from the original on April 11, 2021.
  2. ^ Nedelman, J and Wallenius, T., 1986. Bernoulli trials, Poisson trials, surprising variances, and Jensen’s Inequality. The American Statistician, 40(4):286–289.
  3. ^ Feller, W. 1968. An introduction to probability theory and its applications (Vol. 1, 3rd ed.). New York: John Wiley.
  4. ^ Johnson, N. L. and Kotz, S. 1969. Discrete distributions. New York: John Wiley
  5. ^ Kendall, M. and Stuart, A. 1977. The advanced theory of statistics. New York: Macmillan.
  6. ^ Drezner, Zvi; Farnum, Nicholas (1993). "A generalized binomial distribution". Communications in Statistics - Theory and Methods. 22 (11): 3051–3063. doi:10.1080/03610929308831202. ISSN 0361-0926.
  7. ^ Hoeffding, W. 1956. On the distribution of the number of successes in independent trials. Annals of Mathematical Statistics (27):713–721.