A Note on Poisson Approximation for Independent Geometric Random Variables

Transcription

1 International Mathematical Forum, 4, 2009, no, A Note on Poisson Approximation for Independent Geometric Random Variables K Teerapabolarn Department of Mathematics, Faculty of Science Burapha University, Chonburi 203, Thailand anint@buuacth Abstract Let W be a sum of n independent geometric random variables In 2007, Teerapabolarn and Wongasem [4] used the Stein-Chen method to give a non-uniform bound in approximating the distribution function of W by the Poisson distribution function with mean λ E(W ) n q ip i, where q i p i In this paper, a non-uniform bound on the such approximation has been given for the Poisson mean λ n q i Mathematics Subject Classification: 60F05, 60G50 Keywords: Distribution function; geometric random variable; negative binomial distribution; Poisson approximation; Stein-Chen method Introduction and main results The geometric random variable X with parameter p (0, ) has probabilities P(X ) q p, q p, 0,,, and qp and qp 2 are its mean and variance respectively Let X,, X n be n independent geometric random variables with P(X i ) qi p i, 0,,, and let W X i If ( ) n + p i s are identical to p, then P(W ) q p n, 0,,, is the negative binomial distribution with parameters n and p It is well nown that if all q i are small, the distribution of W can be approximated by the Poisson distribution with mean λ E(W ) n q ip i Correspondingly, the distribution function of W can also be approximated by the Poisson distribution function with the same mean In 2008, Teerapabolarn and Wongasem [4] used the Stein-Chen method to give a non-uniform bound for the difference of the

2 532 K Teerapabolarn distribution function of W and the Poisson distribution function as follows: P(W w 0) λ (e λ ) min, qi 2 p i, p i () () and, if p i s are identical to p, then () becomes P(W w 0) (eλ ) min, q, (2) p() where w 0 N 0 This paper uses the Stein-Chen method to give a non-uniform error bound for approximating the distribution function of W by the Poisson distribution function with mean λ n q i The following theorem and corollary are our main results Theorem 2 Let w 0 N 0 and λ n q i, then λ (e λ ) min, qi 2 P(W w 0 ) p i () Corollary 2 If p i s are identical to p, then (e λ ) min, p() q P(W w 0 ) 0 (3) 0 (4) Remar Since e nq <e nqp, (e nq ) > (e nqp ) Thus the bound (4) is certainly better than the bound (2) 2 Proof of main results We will prove our main results by using the Stein-Chen method The method was originally formulated for normal approximation by Stein [3], and it was adapted and applied to the Poisson case by Chen [2] This method started by Stein s equation for Poisson distribution with a parameter λ, which, given h, is defined by λf(w +) wf(w) h(w) P λ (h), (2) where P λ (h) e λ defined on N 0 h() λ and f and h are bounded real valued functions

3 A note on Poisson approximation 533 For w 0 N 0, let h w0 : N 0 R be defined by if w w 0, h w0 (w) 0 if w>w 0 (22) Then, following [] on pp 7, the solution f(w) of (2) can be expressed in the form (w )!λ w e λ [P λ (h w0 )P λ ( h w )] if w 0 <w, f(w) (w )!λ w e λ [P λ (h w )P λ ( h w0 )] if w 0 w, (23) 0 if w 0 The following lemma is established for proving the main results Lemma 2 Let w 0 N 0 and N \ Then we have 0 < sup f(w) λ (e λ ) min w, (24) Proof Since f(w) > 0 for every w>0, it suffices to show that sup f(w) λ (e λ ) min w, For w w 0, we have f(w) (w )!λ w e λ P λ ( h w0 ) λ j w (w )! j! j λ (w 0 +) w (w )! (w 0 + )! + λ(w0+2) w (w 0 + 2)! + (w )!λ(w 0+) w (w 0 + )! λ (w 0+) w + (w )!λ(w 0+2) w (w 0 + 2)! + () ( w 0 ) [(w0 +) w]! + λ (w 0+2) w w (w 0 +2) ( w 0 ) + [(w0 +2) w]! + w λ 2! + λ2 3! + λ λ 2 2! + λ3 3! + λ (e λ ),

4 534 K Teerapabolarn and we also obtain f(w) λ (e λ ) For w w 0 + and w, we have λ j w f(w) (w )! j! jw (w )! w! + λ + λ this implies that (w )! w! λ w f(w) λ (e λ ) Hence, the inequality (24) is proved λ + λ2 2! + (w + )! + w + +, and f(w) λ (e λ ) Proof of Theorem From (2), given h h w0, we have P(W w 0 ) E[λf(W +) Wf(W )] E[q i f(w +) X i f(w )], (25) where f is defined as in (23) Let W i W X i Then, for each i, we get E[q i f(w +) X i f(w )] E[q i f(w i + X i +) X i f(w i + X i )] EE[(q i f(w i + X i +) X i f(w i + X i )) X i ] E [(q i f(w i + X i +) X i f(w i + X i )) X i 0]P(X i 0) + E [(q i f(w i + X i +) X i f(w i + X i )) X i ]P(X i ) + 2 E [(q i f(w i + X i +) X i f(w i + X i )) X i ] P(X i ) E[p i q i f(w i + )] + E[p i q 2 i f(w i +2) p i q i f(w i + )] + 2 E[p i q + i f(w i + +) p i qi f(w i + )]

5 A note on Poisson approximation 535 p i qi 2 E[f(W i + 2)] + E[p i qi + f(w i + +) p i qi f(w i + )] 2 E[p i qi f(w i + ) p i qi f(w i + )] 2 2( )p i q i E[f(W i + )] 2( )p i q i E[f(W i + )] )p i qi 2( sup f(w) w λ (e λ ) min, which gives λ (e λ ) min λ (e λ ) min, ( )p i qi 2 qi 2, p i (), qi 2 p i () E[q if(w +) X i f(w )] 0 (26) Hence, by (25) and (26), the theorem is proved References [] A D Barbour, L Holst, S Janson, Poisson approximation, Oxford Studies in Probability 2, Clarendon Press, Oxford, 992 [2] L H Y Chen, Poisson approximation for dependent trials, Ann Probab, 3(975), [3] C M Stein, A bound for the error in normal approximation to the distribution of a sum of dependent random variables, ProcSixth Bereley Sympos Math Statist Probab, 3(972), [4] K Teerapabolarn and P Wongasem, Poisson approximation for independent geometric random variables, Int Math Forum, 2(2007), Received: October, 2008