A POINTWISE BINOMIAL APPROXIMATION FOR BERNOULLI RANDOM SUMMANDS

The aim of this paper is to give a pointwise bound for the point metric between the distribution of random sums of independent Bernoulli ran- dom variables and a binomial distribution. Two examples have been given to illustrate the result obtained.


Introduction
Let X 1 , X 2 , ... be a sequence of independent Bernoulli random variables, each with probability p i = P (X i = 1) = 1 − P (X i = 0), and let N be a nonnegative integer-valued random variable and independent of the X i 's.Let S N be random sums of N independent Bernoulli random variables, that is, S N = N i=1 X i .For N = n ∈ N is fixed, there have been some research related to the binomial approximation for the sum S n , which can be found in [1], [2] and [3].Especially, [3] gave a pointwise bound for the point metric between the distribution of S n and an appropriate binomial distribution, which is similar to the aim of this study.Let n = E(N ) and p = 1 − q = λ n , where n ∈ N and λ = E(λ N ) = E N i=1 p i .Let s N (x) be the probability function of S N and b n, p (x) the binomial probability function with parameters n and p, where x ∈ {0, ..., n}.In this paper, we are interested to give a pointwise bound on the point metric |s N (x)−b n, p (x)| for x ∈ {0, ..., n}, which is in Section 2. In Section 3, two examples have been given to illustrate the desired result.Conclusion of this study is presented in the last section.

Result
The following theorem presents a pointwise bound for |s where s N (0) = E( N i=1 q i ).Proof.Let p λ (x) be the Poisson probability function with mean λ.It follows the fact that (2.2) Teerapabolarn [4] showed that and Kun and Teerapabolarn [5] showed that Hence, the inequality (2.1) is obtained by taking the bounds in (2.3) and (2.4) into (2.2).
If X i 's are identically distributed, then the following corollary is an immediately consequence of the Theorem 2.1 Corollary 2.1.For n ∈ N and x ∈ {1, ..., n}, if p 1 = p 2 = • • • = p, then we have the following: where s N (0) = E(q N ).

Examples
Two examples are given to illustrate the result in the case of X i 's are identically distributed.

Conclusion
In this study, a pointwise bound for the point metric between the distribution of random sums of independent Bernoulli random variables and an appropriate binomial distribution was obtained.It is pointed out that the binomial probability function with parameters n and p can be used as an estimate of the probability function of random sums of independent Bernoulli random variables when p is small.