Abstract
Butler and Stephens (2017) have investigated the exact and approximate distributions of a sum S of independent binomial random variables with different probabilities. They used a convolution approach to find the exact distribution, whereas they heavily used the moments and cumulants to find approximations. We propose two different approaches. First, we show that the moment generating function (MGF) approach is easier and simpler to implement for finding the exact distribution of S. We also provide approximations for the distribution of S using large-scale computer simulations based on one million independent replications each. Such exact and approximate distributions are very close to analogous values reported in Butler and Stephens (2017). We show versatility of our approaches by including both exact and approximate distributions for the sum S of independent multinomial, geometric and other discrete random variables.
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