The Stein-Chen method is used to derive two formulas of uniform and non-uniform bounds on Poisson approximation for a sum of n independent geometric random variables. Application of these formulas is illustrated with the Poisson approximation to the negative binomial distribution. Mathematics Subject Classification: Primary 60F05

We use the Stein-Chen method to determine a bound on the relative error between the Poisson binomial distribution function with parameter p = (p 1 , ..., p n ) and the Poisson distribution function with mean λ = n i=1 p i 1−p i . With this bound, the Poisson distribution function with this mean can be used as an estimate of the Poisson binomial distribution function whenever all p i are small.

This paper gives an approximation of the beta binomial distribution with parameters n, α and β by an improved binomial distribution with parameters n and α α+β . The improved binomial approximation is more accurate than the binomial approximation when α + β is large.

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