A New Improvement of Negative Binomial Approximation for

Applied Mathematical Sciences, Vol. 8, 2014, no. 80, 3991 - 3995
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.45360
A New Improvement of Negative Binomial
Approximation for the Negative Hypergeometric
Distribution
K. Jaioun and K. Teerapabolarn*
Department of Mathematics, Faculty of Science
Burapha University, Chonburi, 20131, Thailand
*Corresponding author
Copyright © 2014 K. Jaioun and K. Teerapabolarn. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we give a new improved negative binomial distribution with
to approximate the negative hypergeometric
parameters r and p = M
N
distribution with parameters N , M and r , by improving the improved negative
binomial distribution in [1]. Some numerical examples are presented to illustrate
the result obtained.
Keywords: negative binomial approximation, negative binomial probability
function, negative hypergeometric probability function.
1 Introduction
Let a box contains N items of which M are defective and N − M are
non-defective. Items are inspected at random without replacement, one at a time.
Let X be the number of trials needed to get the first r defective items, Then
X has a negative hypergeometric distribution and its probability function,
nh ( x; r , N , M ) , can be expressed as
3992
K. Jaioun and K. Teerapabolarn
⎛ x − 1⎞ ⎛ N − x ⎞
⎜
⎟⎜
⎟
r −1⎠ ⎝ M − r ⎠
⎝
nb ( x; r , N , M ) =
, x = r , r + 1,..., N − M + r ,
⎛N⎞
⎜ ⎟
⎝M ⎠
where N , M ∈ ` and r ∈ {1, 2,..., M } . The mean and variance of
μ=
(1.1)
X are
r ( N + 1)
r ( N + 1)( N − M )( M + 1 − r )
and σ 2 =
, respectively. From [2], it
2
M +1
( M + 1) ( M + 2)
follows that if N → ∞ and p =
M
remains a constant, then nh ( x ; r , N , M )
N
⎛ x − 1⎞ r x − r
→ nb( x ; r , p ) = ⎜
⎟ p q , q = 1 − p, for every x ∈ {r , r + 1,..., N − M + r} .
⎝ r −1⎠
Therefore, the negative binomial distribution with parameters r and p can be used
as an estimate of the negative hypergeometric distribution with parameters N, M
and r when N is large. In this case, Dongping Hu et al. [1] gave an improved
negative binomial approximation for the negative hypergeometric probability
function as follows:
m ( x ; r, p )
nh ( x ; r , N , M ) ≈ nb
⎧ x( x − 1)
⎫
1
= nb ( x ; r , p ) ⎨1 +
−
⎡⎣( x − r )( x − r − 1) p + r ( r − 1) q ⎤⎦ ⎬ (1.2)
2N
2 Npq
⎩
⎭
where x ∈ {r , r + 1,..., N − M + r} .
Although the improved approximation in (1.2) gives a good approximation
for the negative hypergeometric distribution, but it does not satisfy the important
m ( x ; r , p ) is not always non-negative
property of probability function, i.e., nb
number for every x ∈ {r , r + 1,..., N − M + r} . For example, when N = 100,
m ( x ; r , p ) < 0 for every
M = 70, r = 3 and p = M / N = 0.7 , we have that nb
m ( x ; r , p ) in (1.2) is inappropriate to
x ≥ 16 . Hence, for some N, M and r, nb
approximate the negative hypergeometric distribution.
In this paper, we are interest to give a new improved negative binomial
approximation for the negative hypergeometric probability function. The accuracy
m ( x ; r, p )
of the approximation is measured in terms of nh ( x ; r , N , M ) − nb
for x ∈ {r , r + 1,..., N − M + r} . The result of this study is in Section 2. In Section 3,
some numerical examples have been given to illustrate this improved
approximation, and the conclusion of this study is presented in the last section.
A new improvement of negative binomial approximation
3993
2 Result
The following lemma directly follows from [1].
Lemma 2.1. For 0 < p = 1 − q < 1 , then
x −1
⎛
i ⎞
∏ ⎜⎝ p − N ⎟⎠ = p x −
i =0
1
x −1
i ⎞
⎛
∏ ⎜⎝1 − N ⎟⎠
i =0
x −1
⎛
= 1+
x ( x − 1) x −1
⎛ 1 ⎞
p + O⎜ 2 ⎟,
2N
⎝N ⎠
(2.1)
x( x − 1)
⎛ 1 ⎞
+ O⎜ 2 ⎟,
2N
⎝N ⎠
(2.2)
i ⎞
1
.
⎠ 1 + x( x − 1) + O ⎛ 1 ⎞
⎜ 2⎟
2 Nq
⎝N ⎠
∏ ⎜1 − Nq ⎟ =
i =0 ⎝
(2.3)
The following theorem shows the result of improved negative binomial
approximation to the negative hypergeometric distribution.
Theorem 2.1.
p=
For
x ∈ {r , r + 1,..., N − M + r} ,
if
r≤
2MN
N −M
and
M
, then we have the following:
N
m ( x ; r, p ) + O ⎛ 1 ⎞
nh ( x ; r , N , M ) = nb
⎜ 2⎟
⎝N ⎠
(2.4)
and for the large N,
m ( x ; r, p ) ,
nh ( x ; r , N , M ) ≈ nb
(2.5)
m ( x ; r , p ) = nb ( x ; r , p ) ⎧1 + x( x − 1) − r ( r − 1) ⎫ ⎧1 + ( x − r )( x − r + 1) ⎫ .
where nb
⎨
⎬ ⎨
⎬
2N
2M ⎭ ⎩
2( N − M ) ⎭
⎩
Proof. It follows from (1.1) that
⎛ x − 1⎞ [ M ...( M − r + 1) ] ⎡⎣( N − M )... ( N − x − M + r + 1) ⎤⎦
nh ( x; r , N , M ) = ⎜
⎟
N ( N − 1)( N − 2)...( N − x + 1)
⎝ r −1⎠
⎡ M ⎛ M r − 1 ⎞ ⎤ ⎡⎛ M ⎞ ⎛ M x − r − 1 ⎞ ⎤
... ⎜ −
⎟ ⎜1 − ⎟ ... ⎜ 1 − −
⎟
⎛ x − 1⎞ ⎢⎣ N ⎝ N
N ⎠ ⎥⎦ ⎢⎣⎝
N⎠ ⎝
N
N ⎠ ⎥⎦
=⎜
⎟
1 ⎞⎛
2 ⎞ ⎛ x −1 ⎞
⎛
⎝ r −1⎠
⎜ 1 − ⎟⎜ 1 − ⎟ ... ⎜ 1 −
⎟
N ⎠
⎝ N ⎠⎝ N ⎠ ⎝
3994
K. Jaioun and K. Teerapabolarn
r −1
i ⎞ x − r −1 ⎛
i ⎞
⎛
∏
⎜ p− ⎟ ∏ ⎜q− ⎟
N ⎠ i =0 ⎝
N⎠
⎛ x − 1 ⎞ i =0 ⎝
=⎜
⎟
x
−
1
i ⎞
⎛
⎝ r −1⎠
∏ ⎜⎝1 − N ⎟⎠
i =0
Using Lemma 2.1, we obtain
⎛
⎞
⎜
⎟
−
x
1
−
r
r
1
⎛
⎞
⎛
⎞ r x−r
( ) +O⎛ 1 ⎞ ⎜
1
⎟
nh ( x; r , N , M ) = ⎜
p
q
1
−
⎜
⎟
⎜ 2 ⎟⎟
2 Np
⎝ N ⎠ ⎠ ⎜ 1 + ( x − r )( x − r − 1) + O ⎛ 1 ⎞ ⎟
⎝ r −1⎠
⎝
⎜ 2 ⎟⎟
⎜
2 Nq
⎝ N ⎠⎠
⎝
⎛ x( x − 1)
⎛ 1 ⎞⎞
× ⎜1 +
+O⎜ 2 ⎟ ⎟
2
N
⎝N ⎠⎠
⎝
=
⎧ x( x − 1) r ( r − 1) ⎫
nb ( x; r , p )
⎛ 1 ⎞
−
+ O⎜ 2 ⎟
1+
⎨
⎬
( x − r )( x − r + 1) ⎩
2N
2M ⎭
⎝N ⎠
1+
2( N − M )
m ( x; r , p ) + O ⎛ 1 ⎞
= nb
⎜ 2⎟
⎝N ⎠
⎛ 1 ⎞
m ( x; r , p )
Also, if N is large, then O ⎜ 2 ⎟ ≈ 0 . So nh ( x; r , N , M ) ≈ nb
⎝N ⎠
3 Numerical examples
The following numerical examples are given to illustrate how well the improved
negative binomial distribution approximates the negative hypergeometric
distribution.
3.1 Let N = 50, M = 30, r = 5, p =
M
N
= 0.6 and the numerical results are as
follows:
x
nh ( x; r , N , M )
m ( x; r , p )
nb
nb ( x; r , p )
5
6
7
8
9
10
11
12
13
0.06725915
0.14946478
0.19362483
0.18912193
0.15309870
0.10754250
0.06721406
0.03791563
0.01945670
0.06739200
0.15033600
0.19314103
0.18579456
0.14863565
0.10478813
0.06752305
0.04074378
0.02340374
0.07776000
0.15552000
0.18662400
0.17418240
0.13934592
0.10032906
0.06688604
0.04204265
0.02522559
m ( x; r , p )
nh ( x; r , N , M ) − nb
0.00013285
0.00087122
0.00048380
0.00332737
0.00446306
0.00275437
0.00030899
0.00282815
0.00394704
nh ( x; r , N , M ) − nb ( x; r , p )
0.01050085
0.00605522
0.00700083
0.01493953
0.01375278
0.00721344
0.00032802
0.00412703
0.00576889
A new improvement of negative binomial approximation
3.2 Let N = 100, M = 80, r = 10, p =
M
N
3995
= 0.8 and the numerical results are
as follows:
x
nh ( x; r , N , M )
m ( x; r , p )
nb
nb ( x; r , p )
10
11
12
13
14
15
16
17
18
19
20
0.09511627
0.21136949
0.24818104
0.20305721
0.12895300
0.06717552
0.02963626
0.01129000
0.00375767
0.00109980
0.00028378
0.09529459
0.21206401
0.24690949
0.20007078
0.12732431
0.06821496
0.03218305
0.01378640
0.00547808
0.00205003
0.00073053
0.10737418
0.21474836
0.23622320
0.18897856
0.12283606
0.06878820
0.03439410
0.01572302
0.00668228
0.00267291
0.00101571
m ( x; r , p )
nh ( x; r , N , M ) − nb
0.00017831
0.00069452
0.00127155
0.00298643
0.00162869
0.00103945
0.00254679
0.00249640
0.00172041
0.00095022
0.00044675
nh ( x; r , N , M ) − nb ( x; r , p )
0.01225791
0.00337887
0.01195783
0.01407865
0.00611693
0.00161268
0.00475784
0.00443301
0.00292462
0.00157311
0.00073193
For approximating the negative hypergeometric distribution in the examples
3.1 and 3.2. When N is large, the improved negative binomial distribution is
more accurate than the negative binomial approximation.
4 Conclusion
This study gave a new improved negative binomial distribution to
approximate the negative hypergeometric distribution with parameters N, M and r.
The new improved probability function was corrected to accord the property of
m ( x ; r , p ) ≥ 0 for every x. In addition, the improved
probability function, nb
negative binomial distribution can be used as an approximation of the negative
hypergeometric distribution, and it also gives a good approximation for the
negative hypergeometric distribution when N is large.
References
[1] D.P. Hu, Y.Q. Cui, A.H. Yin, An Improved Negative Binomial
Approximation for Negative Hypergeometric Distribution, Applied
Mechanics and Materials, 427-429 (2013), 2549-2553.
[2] N.L. Johnson, S. Kotz, A.W. Kemp, Univariate Discrete Distributions, 3rd
ed., Wiley, New York, 2005.
Received: May 21, 2014

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