International Journal of Pure and Applied Mathematics Volume 9 No. 3 204, 369-373 ISSN: 3-8080 printed version); ISSN: 34-3395 on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v9i3.9 PAijpam.eu AN IMPROVED POISSON TO APPROXIMATE THE NEGATIVE BINOMIAL DISTRIBUTION K. Teerapabolarn Department of Mathematics Faculty of Science Burapha University Chonburi, 203, THAILAND Abstract: This paper gives an improved Poisson distribution with mean λ for approximating the negative binomial distribution with parameters n and p, where λ = nq = n p). The improved approximation is more appropriate than the well-known Poisson approximation when n is sufficiently large and q is sufficiently small. AMS Subject Classification: 62E7, 60F05 Key Words: negative binomial probability function, Poisson approximation, Poisson probability function. Introduction The negative binomial distribution is an important discrete distribution as same as other discrete distributions. Its applications appear in fields such as automobile insurance, inventory analysis, telecommunications networks analysis and population genetics. Let X be the negative binomial random variable with pa- Received: November 22, 203 c 204 Academic Publications, Ltd. url: www.acadpubl.eu
370 K. Teerapabolarn rameters n > 0 and p 0,), then the probability function in our attention is of the form nb n,p x) = Γn+x) q x p n, x = 0,,...,.) Γn)x! and the mean and variance of X are EX) = nq nq p and variance VarX) = p, 2 respectively. Under parametrization, λ = nq and p = n λ n, it can be expressed as nb n,p x) = λx Γn+x) x! Γn)n x n) λ n, x = 0,,....2) Observe that if n and q 0 while λ = nq remains fixed, then nb n,p x) p λ x) = e λ λ x x! for every x N {0}. Therefore, the Poisson probability function with mean λ = nq can be used as an estimate of the negative binomial probability function if n is large and q is small. In this case, Teerapabolarn [2] gave a non-uniform bound on nb n,p x) p λ x) for x N {0}. In this paper, we are interested to determine an improved Poisson probability function, p λ x), for approximating the negative binomial probability function, and the accuracy of the approximation is measured in the form of nb n,p x) p λ x) for x N {0}. The result of this study is in Section 2. In Section 3, some numerical examples are given to illustrate the improved approximation and the conclusion of this study is presented in the last section. 2. Result Before giving an improved Poisson distribution, we also need the following lemma, which similar to that of []. Lemma 2.. For x N and n > 0, then x + O ) n. Assum- 2 n + O ) n. Thus, ) 2 = {+ kk ) +O ) } n + k n) = 2 Proof. For x =, ing x = k for k N such that k for x = k +, we have k + i ) = + xx ) ) +O n n 2. 2.) ) + i n = = + ) ) + i = + kk ) + i n
AN IMPROVED POISSON TO APPROXIMATE... 37 + kk ) + k n + O ) n = + k+)k 2 + O ) n. Therefore, by mathematical induction, 2.) holds. 2 { Theorem 2.. Letx N {0}, λ = nqand p λ x) = p λ x)e λ p n Then we have the following: + xx ) }. ) nb n,p x) = p λ x)+o n 2 2.2) and for large n and small q, p λ x) = nb n,p x). 2.3) Proof. For x = 0, it is clear that nb n,p 0) = p n = p λ 0)+O n 2 ). Next, we have to show that 2.2) holds for x N. Using.2), we obtain nb n,p x) = λx x! pn x + i ) n { = p λ x)e λ p n + xx ) = p λ x)+o n 2 ). )} +O n 2 by 2.)) Also, if n is large and q is small, then O n 2 ) = 0. Hence pλ x) = nb n,p x). 3. Numerical Examples The following examples are given to illustrate how well the improved Poisson distribution with mean λ = nq approximates the negative binomial distribution with parameters n and p when n is sufficiently large and q is sufficiently small). 3.. Let n = 30 and p = 0.9, then λ = 3.0 and the numerical results are as follows:
372 K. Teerapabolarn x nb n,px) p λ x) p λ x) nb n,px) p λ x) nb n,px) p λ x) 0 0.042396 0.042396 0.04978707 0.00000000 0.0073959 0.277347 0.277347 0.49362 0.00000000 0.0228773 2 0.97889 0.97889 0.224048 0.00000000 0.02692292 3 0.202605 0.20983623 0.224048 0.0004239 0.037866 4 0.7346462 0.76849 0.680336 0.0078043 0.00543326 5 0.795594 0.44563 0.00888 0.0034998 0.07373 6 0.06880763 0.0643857 0.0504094 0.00442606 0.0839823 7 0.03538678 0.032705 0.0260403 0.004573 0.0378275 8 0.0636639 0.033368 0.00805 0.00303020 0.00826488 9 0.0069025 0.00505855 0.00270050 0.008570 0.00420975 0 0.00269500 0.007245 0.000805 0.00097049 0.0088485 0.00098000 0.00053303 0.00022095 0.00044697 0.00075905 2 0.00033483 0.0005050 0.00005524 0.0008433 0.00027960 3 0.000088 0.00003907 0.0000275 0.0000690 0.00009543 3.2. Let n = 00 and p = 0.95, then λ = 5.0 and the numerical results are as follows: x nb n,px) p λ x) p λ x) nb n,px) p λ x) nb n,px) p λ x) 0 0.00592053 0.00592053 0.00673795 0.00000000 0.0008742 0.02960265 0.02960265 0.03368973 0.00000000 0.00408709 2 0.07474668 0.07474668 0.08422434 0.00000000 0.00947766 3 0.2706936 0.2704469 0.4037390 0.00002467 0.0330454 4 0.636080 0.634328 0.7546737 0.0007052 0.086557 5 0.704587 0.6959849 0.7546737 0.00054738 0.0053250 6 0.4887764 0.4775626 0.462228 0.00237 0.00265483 7 0.27264 0.04663 0.0444486 0.006750 0.00827678 8 0.07538260 0.0734926 0.06527804 0.0096334 0.000456 9 0.04522956 0.04333776 0.03626558 0.008980 0.00896398 0 0.024650 0.0230285 0.083279 0.0054726 0.0065732 0.0232505 0.022552 0.0082428 0.0009953 0.00408288 2 0.00570034 0.00500924 0.00343424 0.000690 0.0022660 3 0.00245553 0.00206590 0.0032086 0.00038963 0.003467 4 0.00099098 0.000797 0.0004774 0.0009927 0.0005925 5 0.00037657 0.00028325 0.0005725 0.00009333 0.0002933 6 0.0003533 0.00009499 0.0000494 0.00004034 0.0000869 For approximating the negative binomial distribution in the examples 3. and 3.2, it can be seen that the improved Poisson distribution is more appropriate than the Poisson distribution. 4. Conclusion In this study, an improved Poisson distribution with mean λ = nq was obtained by using some mathematical manipulations. This improved approximation is more accurate than the well-known Poisson approximation, thus the improved Poisson distribution can also be used as an estimate of the negative binomial distribution when n is sufficiently large and q is sufficient small.
AN IMPROVED POISSON TO APPROXIMATE... 373 References [] D.P. Hu, Y.Q. Cui, A.H. Yin, An improved negative binomial approximation for negative hypergeometric distribution, Applied Mechanics and Materials, 427-429 203), 2549 2553. [2] K. Teerapabolarn, A pointwise approximation for independent geometric random variables, International Journal of Pure and Applied Mathematics, 76 202), 727 732.
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