An Improved Geometric Approximation for a Beta Binomial

Applied Mathematical Sciences, Vol. 8, 2014, no. 161, 8037 - 8040
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.410809
An Improved Geometric Approximation
for a Beta Binomial Distribution
K. Teerapabolarn
Department of Mathematics, Faculty of Science
Burapha University, Chonburi 20131, Thailand
c 2014 K. Teerapabolarn. This is an open access article distributed under
Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Abstract
The aim of this paper is to give an approximation of a beta binomial
distribution with parameters n and β by an improved geometric distriβ
. The improved geometric approximation is
bution with parameter β+n
more accurate than the geometric approximation when β is large.
Mathematics Subject Classification: 62E17, 60F05
Keywords: Beta binomial distribution, Geometric distribution, Improved
geometric distribution, Probability function
1
Introduction
The beta binomial distribution with parameters n ∈ N and β > 0 (α = 1 is
fixed) is the binomial distribution with parameter n and a random variable p
that has a beta distribution with shape parameters α = 1 and β > 0. Let X
be this beta binomial random variable with the probability function as follows:
bbn,β (x) =
n!Γ(β + n − x)β
, x = 0, 1, ..., n.
(n − x)!Γ(β + n + 1)
(1.1)
nβ(n+β+1)
n
The mean and variance of X are µ = β+1
and σ 2 = (β+1)
2 (β+2) , respectively.
n
It is observed that β → ∞ while q = 1 − p = β+n
remains a constant,
then bbn,β (x) → gp (x) = q x p for every x ∈ {0, ..., n}. In this paper, we are
8038
K. Teerapabolarn
bp (x), to
interest to determine an improved geometric probability function, g
approximate the beta binomial probability function in (1.1). The accuracy
bp (x)| for x ∈
of the approximation is measured in the form of |bbn,β (x) − g
{0, 1, ..., n}, which is in Section 2. In Section 3, some numerical examples have
been given to illustrate the improved approximation and the conclusion of this
study is presented in the last section.
2
Result
Using the same idea from in [1], the following lemma can be also obtained.
Lemma 2.1. For β ∈ R+ , x, n ∈ N and 0 < q < 1, then
x−1
Y
qx
i
,
=
q−
x(x−1)
β
+
n
1 + 2(β+n)q + O β12
i=0
1
x(x + 1)
1
=1+
+O
.
Qx i
2(β + n)
β2
1
−
i=1
β+n
Theorem 2.1. For x ∈ {0, 1, ..., n} and p =
(2.1)
(2.2)
β
,
β+n
we have
1
bp (x) + O
bbn,β (x) = g
,
β2
bp (x)
bbn,β (x) ≈ g
(2.3)
(2.4)
x(x+1)
bp (x) =
for large β, where g
q x p{1+ 2(β+n) }
1+
x(x−1)
2n
.
Proof. Applying Lemma 2.1, it follows that
n!β
(n − x)!(β + n) · · · (β + n − x)
Qx−1 i
q
−
p
i=0
β+n
= Q x
i
i=1 1 − β+n
n
o
x(x+1)
q x p 1 + 2(β+n)
1
=
+O
x(x−1)
β2
1 + 2n
1
bp (x) + O
=g
.
β2
bbn,β (x) =
If β is large, then O
1
β2
bp (x).
≈ 0. Hence bbn,β (x) ≈ g
An improved geometric approximation for a beta binomial distribution
3
8039
Numerical examples
The following examples are given to illustrate how well an improved geometric
distribution approximates a beta binomial distribution.
3.1. Let n = 20 and β = 130, then p =
as follows:
x
0
1
2
3
4
5
6
7
8
9
bbn,β (x)
0.86666667
0.11633110
0.01493440
0.00182870
0.00021293
0.00002350
0.00000245
0.00000024
0.00000002
0.00000000
bp (x)
g
0.86666667
0.11632593
0.01496720
0.00185782
0.00022475
0.00002678
0.00000317
0.00000038
0.00000004
0.00000001
gp (x)
0.86666667
0.11555556
0.01540741
0.00205432
0.00027391
0.00003652
0.00000487
0.00000065
0.00000009
0.00000001
bbn,β (x)
0.78571429
0.16957862
0.03563609
0.00728329
0.00144595
0.00027848
0.00005195
0.00000938
0.00000163
0.00000027
0.00000004
0.00000001
0.00000000
bp (x)
g
0.78571429
0.16956997
0.03566306
0.00732954
0.00147917
0.00029478
0.00005832
0.00001151
0.00000227
0.00000045
0.00000009
0.00000002
0.00000000
gp (x)
0.78571429
0.16836735
0.03607872
0.00773115
0.00165668
0.00035500
0.00007607
0.00001630
0.00000349
0.00000075
0.00000016
0.00000003
0.00000001
and the numerical results are
bbn,β (x) − g
bp (x)
0.00000000
0.00000517
0.00003280
0.00002912
0.00001182
0.00000329
0.00000072
0.00000014
0.00000002
0.00000000
3.2. Let n = 30 and β = 110, then p =
as follows:
x
0
1
2
3
4
5
6
7
8
9
10
11
12
130
150
110
140
bbn,β (x) − gp (x)
0.00000000
0.00077554
0.00047301
0.00022562
0.00006098
0.00001303
0.00000242
0.00000041
0.00000006
0.00000001
and the numerical results are
bbn,β (x) − g
bp (x)
0.00000000
0.00000865
0.00002698
0.00004625
0.00003323
0.00001630
0.00000637
0.00000213
0.00000064
0.00000018
0.00000005
0.00000001
0.00000000
bbn,β (x) − gp (x)
0.00000000
0.00121128
0.00044263
0.00044787
0.00021073
0.00007652
0.00002412
0.00000693
0.00000186
0.00000047
0.00000012
0.00000003
0.00000001
From the examples 3.1 and 3.2, it can be seen that the improved geometric
approximation is more accurate than the geometric approximation.
4
Conclusion
This study gave an approximation of a beta binomial distribution with parameters n and β (α = 1) by an improved geometric distribution with parameter
β
. By numerical comparison, the improved geometric approximation is more
β+n
accurate than the geometric approximation when β is sufficiently large, that
is, the beta binomial distribution with parameters n and β can be well approxβ
imated by an improved geometric distribution with parameter β+n
when β is
sufficiently large.
8040
K. Teerapabolarn
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.
http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.2549
Received: October 15, 2014; Published: November 19, 2014

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