eu ON BINOMIAL APPROXIMATION FOR INDEPENDENT BERNOULLI RANDOM SUMS

A non-uniform bound for the distance between the distribution function of random sums of independent Bernoulli random variables and an appropriate binomial distribution function could be obtained. 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 of success p i = P (X i = 1) = 1 − P (X i = 0), and let N be a non-negative integer-valued random variable and independent of the X i 's.Thus S N = N i=1 X i is the random sums of N independent Bernoulli random variables.In the case of N = n ∈ N is fixed, there have been some research on topic related to the binomial approximation to the distribution of the sum S n , which can be found in [2], [3] and [5].Let λ N = N i=1 p i , n = E(N ) ∈ N and p = 1 − q = λ n , where λ = E(λ N ).For x ∈ {0, 1, ..., n}, let S n (x) = P (S n ≤ x) and B n,p (x) = P (B n,p ≤ x), where B n,p is the binomial random variable with parameters n and p.In this paper, we are interested to give a non-uniform bound for the distance between S N (x) and url: www.acadpubl.eu is in Section 2. In Section 3, two examples have been given to illustrate the desired result.The conclusion of this study is presented in the last section.

Result
The following theorem presents a non-uniform bound for the distance between the distribution functions of S N and B n, p , which is the desired result.
Proof.Let P λ (x) be the Poisson distribution function with mean λ at x.It follows the fact that Following [6], we have Applying Theorem 1.C in [1] together with Lemma 2.1 in [5] and using Corollary 4.1 in [5], we also obtain and respectively.Hence, the inequality (1) is obtained by taking the bounds in (3), ( 4) and ( 5) into (2).
If X i 's are identically distributed, then the following corollary is an immediately consequence of the Theorem 2.1.

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

Conclusion
In this study, a non-uniform bound for the distance between the distribution function of random sums of independent Bernoulli random variables and an appropriate binomial distribution function with parameters n and p cloud be obtained.The bound points out that the distribution function of random sums of independent Bernoulli random variables can be approximated by the binomial distribution function with parameters n and p when p is small, that is, the desired result gives a good binomial approximation when p is small.