Approximating the number of successes in independent trials: binomial versus Poisson
The authors consider approximations to the distribution of a sum \(\sum^n_{i=1}\) of independent Bernoulli random variables \(X_i\sim \text{Be}(p_i)\) by binomial and Poisson random distributions with the same mean \(\lambda= \sum^n_{i=1} p_i\). They show that total variation approximation is better using the binomial distribution \(\text{Bi}(k,\lambda/k)\) than that using \(\text{Bi}(m,\lambda/m)\) for any \(m> k\), including the Poisson case corresponding to \(m=\infty\), for \(k\) given explicitly in terms of the parameters of the problem; frequently, \(k=n\). They conjecture that the result is actually true for another, smaller value of \(k\).
scientific article; zbMATH DE number 1906159