The Distribution of a Sum of Independent Binomial Random Variables

The distribution of a sum S of independent binomial random variables, each with different success probabilities, is discussed. An efficient algorithm is given to calculate the exact distribution by convolution. Two approximations are examined, one based on a method of Kolmogorov, and another based on fitting a distribution from the Pearson family. The Kolmogorov approximation is given as an algorithm, with a worked example. The Kolmogorov and Pearson approximations are compared for several given sets of binomials with different sample sizes and probabilities. Other methods of approximation are discussed and some compared numerically. The Kolmogorov approximation is found to be extremely accurate, and the Pearson curve approximation useful if extreme accuracy is not required.

[1]  R. Eisinga,et al.  Saddlepoint approximations for the sum of independent non‐identically distributed binomial random variables , 2013 .

[2]  Raaj Kumar Sah.  Comparative properties of sums of independent binomials with different parameters , 1989 .

[3]  Michael V. Boutsikas,et al.  A bound for the distribution of the sum of discrete associated or negatively associated random variables , 2001 .

[4]  J. Arthur Woodward,et al.  On the exact convolution of discrete random variables , 1997 .

[5]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[6]  J. Sunklodas.  On the normal approximation of a binomial random sum , 2014 .

[7]  Peter D. Welch,et al.  The Fast Fourier Transform and Its Applications , 1969 .

[8]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[9]  F. Proschan,et al.  The Reliability of $K$ Out of $N$ Systems , 1983 .

[10]  Robert J. Elliott,et al.  Binomial Models in Finance , 2005 .

[11]  A Pólya Approximation to the Poisson-Binomial Law , 2012, Journal of Applied Probability.

[12]  E. S. Pearson.  Biometrika tables for statisticians , 1967 .

[13]  James C Benneyan,et al.  Exact and approximate probability distributions of evidence-based bundle composite compliance measures , 2010, Health care management science.

[14]  Herbert Solomon,et al.  Approximations to Density Functions Using Pearson Curves , 1978 .

[15]  R. G. Sharafiev,et al.  LITERATURE CITED N. V. smirnov and I. V. Dunin-Barkovskii, Course in Probability Theory and Mathematical Statistics for Technical Applications (in Russian), Nauka, Moscow (1965). AGING OF ~TAL OF REACTION VESSELS OF SLOW COKING EQUIPMENT , 1984 .

[17]  Andrew P. Robinson,et al.  Confidence Intervals for the Weighted Sum of Two Independent Binomial Proportions , 2012 .

[18]  M. Stephens,et al.  The Distribution of a Sum of Binomial Random Variables , 1993 .

[19]  H. Solomon,et al.  Distribution of a Sum of Weighted Chi-Square Variables , 1977 .

[20]  Adrian Röllin,et al.  A Three-Parameter Binomial Approximation , 2009, Journal of Applied Probability.