Download Some Mathematical Characteristics of the Beta Density Function of

Some Mathematical Characteristics of the Beta
Density Function of Two Variables
M. Ha…dz Omar and Anwar H. Joarder
Department of Mathematics and Statistics
King Fahd University of Petroleum and Minerals
Dhahran 31261, Saudi Arabia
Emails: [email protected]; [email protected]
Abstract Some well known results on the bivariate beta distribution have been reviewed. Corrected product moments are derived. These moments will be important for studying further characteristics of the distribution. The distribution of the ratio of two correlated beta variables has been
derived and used to obtain a new reliability expression. Other interesting distributions stemming
from the correlated beta variables are also derived.
AMS Mathematics Subject Classi…cation: 60E05, 60E99
Key Words and Phrases: bivariate distribution, central moments, product moments, correlation coe¢ cient, reliability
1
Introduction
The bivariate beta distribution has applications in areas such as voting analysis of political issues
of two competing candidates and research on soil strength (see Hutchinson and Lai, 1990, p104). In
this paper we derive some centered moments that are important in studying further properties of
the distribution. We also review some well known results. In addition, we derive some new results
for which some applications are o¤ered.
The beta distribution is given by
f (x) =
1
xm
B(m; n)
1
(1
x)n
1
;
(1.1)
(m) (n)
.
(m + n)
A distribution is said to be a beta distribution of a second kind BetaII(m; n) if its density
function is given by
where m; n > 0; 0
x
1 and B(m; n) =
f (y) =
where m; n > 0; 0
1
ym 1
;
B(m; n) (1 + y)m+n
(1.2)
y < 1. A transformation Z = 1=(1 + Y ) of the random variable Y in (1.2) will
1
result in a Beta(m; n) distribution. It is interesting to note that if m = n, the density function in
(1.2) is also the density function of an F (2m; 2m) distribution.
The product moments of order a and b for two random variables X and Y are de…ned by
= E X a Y b while the centered product moments (sometimes called central product moments,
corrected moments or central mixed moments) are de…ned by
0
a;b
a;b
= E (X
E(X))a (Y
E(Y ))b :
The former moments 0a;b are often called product moments of order zero or raw product moments.
Evidently 0a;0 = E (X a ) is the a-th moment of X, and 00;b = E Y b is the b-th moment of Y . In
case X and Y are independent 0a;b = E X a )E(Y b = 0a;0 00;b : Interested readers may go through
Johnson et. al. (1993, p46).
The correlation coe¢ cient
( 1<
< 1) between X and Y is denoted by
X;Y
2
Note that 2;0 = E (X E(X)) =
moment, 1;1 = E [(X E(X)) (Y
between X and Y .
=p
1;1
:
(1.3)
2;0 0;2
which is popularly denoted by
E(Y ))] denoted popularly by
2
1
20
12 ,
while the central product
is in fact the covariance
The importance of evaluating central moments of a bivariate distribution cannot be overlooked.
In a series of papers, Mardia (1970a, 1970b, 1970c, 1974, 1975) de…ned and discussed the properties
of moments based on Mahalanobis distance. As it is di¢ cult to derive distribution of Mahalanobis
distance for many distributions, bivariate or multivariate, and calculate moments thereof, Joarder
(2006) derived Mahalanobis moments (or simply, standardized moments) in terms of central product
moments. He showed that the central moments can be used as an alternative way to describe further
important characteristics of a bivariate distribution, for example, bivariate kurtosis coe¢ cient.
It should be mentioned that the central moments derived in Section 4 is a formidable task
requiring meticulous calculation. Some discusssions on the product moment correlation is given in
section 5. In Section 6, the distribution of the ratio of two correlated beta variables has been derived
and used to obtain a new reliability expression. In addition, some other interesting distributions are
also derived.
2
A Review of Some Distributional Properties of the Bivariate Beta Distribution
The bivariate beta is a natural extension of a univariate beta distribution. The probability density
function of the bivariate beta distribution is given by
f (x; y) =
(m + n + p) m
x
(m) (n) (p)
1 n 1
y
(1
x
y)p
1
;
(2.1)
where m; n; p > 0; x
0; y
0; and x + y
1. A natural extension of this bivariate beta distribution is the multivariate Dirichlet distribution (see Fang and Zhang, 1990, p47). Figure 1 in
the Appendix provides a surface graph of the pdf in (2.1) for di¤erent values of m; n; and p. The
following theorem is due to Lee (1971).
Theorem 2.1 Let X and Y have the joint pdf given by (2.1). Then the marginal probability density
2
functions of the bivariate beta distribution with pdf in (2.1) are given by:
(i) X
(ii) Y
Beta(m; n + p);
Beta(n; m + p):
(2.2)
Proof. The marginal pdf of Y follows from the substitution of u (1
R 1 y (m + n + p) m 1 n 1
g(y) = 0
x
y
(1 x y)p 1 dx:
(m) (n) (p)
y) = x in the following integral
Thus Y follows Beta(n; m + p): Similarly, X follows Beta(m; n + p):
We note that the mean and variance of Y are given by
E(Y ) =
n
n(m + p)
; and V (Y ) =
m+n+p
(t + 1)t2
respectively; where t = m + n + p. The mean and variance of X are given by
E(X) =
m(n + p)
m
; and V (X) =
respectively:
m+n+p
(t + 1)t2
Theorem 2.2. Let X and Y have the joint pdf given by (2.1). Then the conditional pdf of Y given
X = x is given by
k(yjx) =
(1 x)
B(n; p)
n 1
y
1
x
1
p 1
y
1
; 0<y<1
x
x; 0 < x < 1:
(2.3)
Proof. The conditional pdf of Y given X = x is de…ned as
f (x; y)=h(x) =
=
(m + n + p) m 1 n 1
x
y
(1 x y)p
(m) (n) (p)
(m + n + p)B(m; n + p) n 1
y
(1 x
(m) (n) (p)
1
xm
B(m; n + p)
1
y)p
1
=(1
x)n+p
1
1
(1
x)n+p
1
1
which can be written as (2.3).
Thus, from (2.3), it can be seen that the conditional distribution of Y =(1 X) given X = x is
Beta(n; p) which implies that E(Y jX = x) = (1 x)n=(n + p) which can also be written as
n
n
x+
n+p
n+p
E(Y j X = x) =
(2.4)
in the regular regression format. Thus, the regression of Y on X is linear. Also
V ar (Y j X = x) = (1
x)2 np=((n + p)2 (n + p + 1))
which is not free from x: This means that the conditional variance for the linear regression of Y on
X is not homoscedastic. The linear regression also suggests that Y is not independent of X. Lee
(1971) also proved that
E(X k j Y = y) = (1
y)k and E(Y k j X = x) = (1
3
x)k :
Theorem 2.3. Let X and Y have the joint pdf given by (2.1). Then Y =(1
pendent.
X) and X are inde-
Proof. Let u = y=(1 x) and v = x with Jacobian J(x; y ! u; v) = (1 v): The support region is
mapped into the region f(u; v) : v > 0; u(1 v) > 0; v + u(1 v) < 1g = f(u; v) : 0 < u < 1; 0 < v < 1g :
Then from (2.1),the joint pdf of U and V is given by
g(u; v) =
That is, U
1
un
B(n; p)B(m; n + p)
Beta(n; p) and V
1 m 1
v
v)n ((1
(1
u)(1
p 1
v))
:
Beta(m; n+p) and they are independently distributed.
Theorem 2.4 Let (X; Y ) follow the bivariate beta distribution with pdf given by (2.1). Also let
U = X + Y and V = X=(X + Y ). Then U Beta(m + n; p) is independent of V
Beta(m; n):
Proof.
Let us make the transformation u = x + y and uv = x . The support region is
mapped onto the region f(u; v) : uv > 0; u(1 v) > 0; u < 1g = f(u; v) : 0 < u < 1; 0 <
v < 1g with Jacobian J(x; y ! u; v) = u. The theorem then follows in a straightforward
manner.
In what follows we will de…ne
a;b
where
= E(X) and
)a (Y
= E (X
)b
(2.5)
= E(Y ):
Provost and Cheong (2000) discussed the distribution of linear combinations of the components
of a Dirichlet random vector. The distribution of ax + by is a special case of that.
3
Raw Product Moments
For any non-negative integer a; Pochhammer factorials are de…ned as
cfag = c(c + 1)(c + 2) (c + a 1)
and cfag = c(c
1)(c
2)
(c
a + 1):
with cf0g = 1; cf0g = 1:
Also, the (a; b)th raw product moment of X and Y of the bivariate beta distribution is given by
E(X a Y b ) =
Z
1
0
Z
1 y
xa y b f (x; y)dxdy:
0
Lemma 3.1. Let X and Y have the joint pdf given by (2.1). Then
(i)
the marginal density function of X Beta(m; n + p), has an expected value of
mfag
;
E(X a ) =
tfag
(ii)
the marginal density function of Y
nfbg
E(Y b ) =
;
tfbg
Beta(n; m + p), has an expected value of
4
(3.1)
(iii)
and the raw product moment of order (a; b) is
mfag nfbg
E(X a Y b ) =
;
where t = m + n + p.
tfa+bg
The above lemma gives rise to some useful raw moments that will be used further in this article.
4
Centered Moments
The centered product moments of a bivariate beta distribution, a;b can be obtained by directly
evaluating
the following integral.
h
i R R
1 1 x
a
b
a
b
E (X1 E(X1 )) (X2 E(X2 )) = 0 0 2 (X1 E(X1 )) (X2 E(X2 )) f (x1 ; x2 )dx1 dx2 :
For illustration, we derive the central moment
1;2
1;2
below.
h
i
1
2
= E (X E(X)) (Y E(Y ))
Z 1Z 1 y
m
n 2
=
x
f (x; y)dxdy
y
t
t
0
0
Z 1Z 1 y
n
n 2
m
nm
n 2m
=
xy 2 2xy + x
y 2 + 2y
f (x; y)dxdy
t
t
t
t t
t
t
0
0
Z 1Z 1 y
Z 1Z 1 y
Z
Z
n
n 2 1 1 y
2
=
xy f (x; y)dxdy 2
xyf (x; y)dxdy +
xf (x; y)dxdy
t 0 0
t
0
0
0
0
Z Z
Z Z
Z Z
mn 1 1 y
n2 m 1 1 y
m 1 1 y 2
y f (x; y)dxdy + 2 2
yf (x; y)dxdy
f (x; y)dxdy
t 0 0
t
t3 0 0
0
0
mn(n + 1)
n
mn
n 2 m
m n(n + 1)
m
n 2 n2 m
=
2
+
+2
t(t + 1)(t + 2)
t t(t + 1)
t
t
t
t(t + 1)
t
t
t3
2
2
2
2
mn(n + 1)
mn
mn
mn(n + 1)
mn
mn
=
2 2
+ 3
+2 3
2
t(t + 1)(t + 2)
t (t + 1)
t
t (t + 1)
t
t3
2 (t 2n) mn
=
:
(t + 1) (t + 2) t3
The higher order moments generally require more meticulous integration and cross-checking of
calculations. More details of these centered moments can be found in Omar and Joarder (2006).
Some centered product moments of order a + b = 2; 3; 4; 5; 6 are given below:
0;2
0;3
0;4
0;5
(t n) n
;
(t + 1) t2
(t 2n) (t n) n
=2
;
(t + 1) (t + 2) t3
t2 (n + 2) n (n + 6) t + 6n2
= 3 (t n) n
;
(t + 1) (t + 2) (t + 3) t4
t2 (5n + 6) n (5n + 12) t + 12n2
= 4 (t 2n) (t n) n
;
(t + 1) (t + 2) (t + 3) (t + 4) t5
=
5
0;6
=
5 (t n) n
(t + 1) (t + 2) (t + 3) (t + 4) (t + 5) t6
[t4 3n2 + 26n + 24
1;1
1;2
1;3
1;4
1;5
2n 3n2 + 56n + 60 t3
+ n2 3n2 + 172n + 240 t2 2n3 (43n + 120) t + 120n4 ];
nm
=
;
(t + 1) t2
(t 2n) mn
;
= 2
(t + 1) (t + 2) t3
mn
=3 4
( (n + 2) t2 + n (n + 6) t 6n2 );
t (t + 1) (t + 2) (t + 3)
mn (t 2n)
12n2 (12 + 5n)nt + (6 + 5n)t2 ;
= 4 5
t (t + 1) (t + 2) (t + 3) (t + 4)
mn
= 5 6
t (t + 1) (t + 2) (t + 3) (t + 4) (t + 5)
[ 3n2 + 26n + 24 t4
2n 3n2 + 56n + 60 t3
+ n2 3n2 + 172n + 240 t2
2;2
2;3
2;4
2n3 (43n + 120) t + 120n4 ];
(m + n) t2 + 3 (2m + 2n + mn) t 18mn
;
(t + 1) (t + 2) (t + 3) t4
t4 t3 (m + 6n) + 12m + 15mn + 5n2 t2 4n (9m + 3n + 5mn) t + 48mn2
= 2mn
;
(t + 1) (t + 2) (t + 3) (t + 4) t5
mn
= 6
t (t + 1) (t + 2) (t + 3) (t + 4) (t + 5)
= mn
t3
[t5 (3n + 6) + t4
6m
40n
3mn
6n2
+ t3 120m + 164mn + 120n2 + 3n3 + 18mn2
+ t2
3;3
480mn
86n3
516mn2
15mn3
+ t 120n3 + 720mn2 + 430mn3
600mn3 ];
mn
=
(t + 1) (t + 2) (t + 3) (t + 4) (t + 5) t
26n2 + m2 (9n + 26) + 9mn (n + 20)
10m + 10n + 3mn
[4 3
+
t
t2
2
2
2
2n (43m + 20) + m 86n + 5n + 40
3
t3
mn (36m + 36n + 43mn)
m2 n2
+ 10
600 5 ]
4
t
t
Similar expressions for moments (b; a) = E (X
2;0
3;0
4;0
5;0
)b (Y
)a are provided below.
(t m) m
;
(t + 1) t2
(t 2m) (t m) m
=2
;
(t + 1) (t + 2) t3
t2 (m + 2) m (m + 6) t + 6m2
= 3 (t m) m
;
(t + 1) (t + 2) (t + 3) t4
t2 (5m + 6) m (5m + 12) t + 12m2
= 4 (t m) (t 2m) m
;
(t + 1) (t + 2) (t + 3) (t + 4) t5
=
6
6;0
=
5 (t m) m
(t + 1) (t + 2) (t + 3) (t + 4) (t + 5) t6
[ 3m2 + 26m + 24 t4
2m 3m2 + 56m + 60 t3
+ m2 3m2 + 172m + 240 t2
2;1
3;1
4;1
5;1
mn (t 2m)
;
t3 (t + 1) (t + 2)
(m + 2) t2 + m (m + 6) t 6m2
= 3mn
;
(t + 1) (t + 2) (t + 3) t4
(5m + 6) t2
12m + 5m2 t + 12m2
= 4 (t 2m) mn
;
(t + 1) (t + 2) (t + 3) (t + 4) t5
mn
= 5 6
t (t + 1) (t + 2) (t + 3) (t + 4) (t + 5)
=
2
[(3m2 + 26m + 24)t4
2m 3m2 + 56m + 60 t3
+ m2 3m2 + 172m + 240 t2
3;2
4;2
2m3 (43m + 120) t + 120m4 ];
2m3 (43m + 120) t + 120m4 ];
t3 (6m + n) + (12n + 5m (m + 3n)) t2 4m (m (5n + 3) + 9n) t + 48m2 n
;
(t + 1) (t + 2) (t + 3) (t + 4) t5
mn
= 6
t (t + 1) (t + 2) (t + 3) (t + 4) (t + 5)
= 2mn
t4
[(3m + 6) t5
(6m2 + m (3n + 40) + 6n)t4
+ (3m3 + 6 (3n + 20) m2 + 4n (41m + 30))t3
m(m2 (15n + 86) + 12n (43m + 40))t2
+ 10m2 (12m + 72n + 43mn) t
5
600m3 n]:
Correlation Coe¢ cient for the Bivariate Beta Distribution
Theorem 5.1 Let X and Y have the joint pdf given by (2.1). Then the product moment correlation
coe¢ cient between X and Y is given by
r
mn
:
(5.1)
=
(n + p) (m + p)
Proof. Substituting the moments from Section 4 in
p
2;0 0;2
=
1;1
we have (5.1).
Figure 2 in the Appendix provides graphs of values of the correlation coe¢ cient for di¤erent values
of p. Mainly, the correlation coe¢ cient reaches the limiting value of -1 when p is small as either m
or n increases.
For a special case of a bivariate beta distribution de…ned in (2.1) where m = n, that is (t = 2m + p);
then the product moment correlation coe¢ cient is = m=(m + p): Note that where m = n; the
two bivariate marginal probability density functions are identical.
Another special case of a bivariate beta distribution de…ned in (2.1) occurs when m = n = p, that is
t = 3m: In this case, the product moment correlation coe¢ cient = 1=2: Note that here we have
a special case of (5.1) where the two bivariate marginal pdfs are identical with m = n = p.
Alternatively, it is also easy to check that
7
Cov(X; Y ) = E(XY )
E(X)E(Y ) =
since E(XY ) = E(XE(Y jX)) = E(X
mn
(t + 1)t2
(1 X)n
mn
)=
.
n+p
t(t + 1)
The above shows that the product moment correlation coe¢ cient
given by what we have (5.1).
6
between X and Y is also
Some New Distribution of Functions of Correlated Beta
Variables and Their Applications to Reliability
Theorem 6.1 Let (X; Y ) follow the bivariate beta distribution with pdf given by (2.1). Then the
pdf of W = Y =X is given by
g(w) =
wn 1
;
B(m; n)(w + 1)m+n
0
w < 1:
(6.1)
Proof. By Theorem 2.4, V = X=(X + Y ) Beta(m; n): With the transformation W = (1 V )=V ,
the support region for bivariate beta distribution is mapped onto the region fw : 0 < w < 1g with
absolute value of Jacobian jJ(v ! w)j = (w + 1) 2 . Then (6.1) follows in a straightforward manner.
Krishnamoorthy et. al. (2007) discussed a special reliability function de…ned as R = P (X > Y ).
This type of reliability function is interesting when one wishes to assess the proportion of time the
random strength variable X of a component exceeds the random stress variable Y which the
component is subjected to. If X Y , then either the component fails or the system the component
supports malfunctions. The same reliability function R can also be used to estimate the probability
that one of the two devices, X and Y , fails before the other. Krishnamoorthy et. al. (2007)
also discussed inference on reliability in a stress-strength relationship that follows a two-parameter
exponential model.
Theorem 6.1 can immediately be applied to the situations where reliability of a component in
a stress-strength relationship follows a bivariate beta distribution. In this situation, the following
Theorem can be used.
Theorem 6.2. Let (X; Y ) follow the bivariate beta distribution with pdf given by (2.1). And let
the pdf of W = Y =X is given by
g(w) =
wn 1
;
B(m; n)(w + 1)m+n
0<w<1
Then, the reliability R = P (X > Y ) de…ned in Krishnamoorthy et. al. (2007) is given by
R = I1=2 (m; n);
where Ia (m; n) =
R 1 (1
a
v)n 1 v m
B(m; n)
(6.2)
1
dv:
Proof. From the de…nition we have R = P (X > Y ) = P (W < 1): But, W = Y =X
n); a Beta distribution of the second kind de…ned in (1.2). So,
8
BetaII(n; m+
R1
wn 1
dw
B(m; n)(w + 1)m+n
Letting W = (1 V )=V;
R 1 (1 v)n 1 v m 1
R = 1=2
dv
B(m; n)
which is symbolically given in the above Theorem.
R=
0
Theorem 6.2 provides an alternative simpler expression of reliability than what is reported in
Nadarajah (2005). Also, the evaluation of reliability is reduced to a univariate integral instead of a
bivariate one.
It is also worth mentioning that with the appropriate transformation that the pdf of S = W 2 =
2
Y =X 2 is given by
m+n
1
1
sn=2 1
;
0 < s < 1:
f (s) =
2B(m; n)
s1=2 + 1
Theorem 6.3. Let (X; Y ) follow the bivariate beta distribution with pdf given by (2.1). And let
X = R2 cos2 and Y = R2 sin2 : Then R2 Beta(m + n; p); sin2
Beta(n; m);
where follows the distribution given by
2
2m 1
2n 1
h( ) =
(cos )
(sin )
:
Beta(m; n)
Proof. Let X and Y follow the bivariate beta probability density function given by (2.1). Also,
let X = R2 cos2 and Y = R2 sin2 : With this transformation, the support region is mapped onto
the region f(r2 ; ) : 0 < r2 < 1; 0 < < =2g with J(x; y ! r2 ; ) = 2r2 cos sin : Substitution into
(2.1) gives the following
(m + n + p)
(m + n)
p 1
m+n
2m 1
2n 1
g(r2 ; ) =
1 r2
r2
2
(cos )
(sin )
(m + n) (p)
(m) (n)
That is, R2 and
are independent random variables with distributions as mentioned in the theorem.
Now, let U = sin2 : Hence with this transformation, the support region is mapped onto the
1=2
region fu : 0 u 1g with J(u ! ) = 2[(1 sin2 )(sin2 )]1=2 = 2 ((1 u)u) :Substituting u
2
into h( ) with the appropriate Jacobian provides the result that U = sin
Beta(n; m):
Corollary 6.1. Let X and Y follow the bivariate beta probability density function given by (2.1).
Also, let X = R2 cos2 and Y = R2 sin2 : If m = n, then R2
Beta(2m; p); U = sin2
2
2m
1
1
Beta(m; m) and h( ) =
sin 2
.
Beta(m; m) 2
Proof. Substituting m = n into Theorem 6.3 provides results of Corollary 6.1:
7
Acknowledgements
The authors gratefully acknowledge the excellent research support provided by King Fahd University of Petroleum & Minerals. The authors would also like to thank the BMMSS editor and two
anonymous reviewers whose suggestions have resulted in this improved version of the manuscript.
9
8
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Appendix
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Figure 1: Bivariate Beta probability density function for various values of m; n; and p.
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Figure 2: Product moment correlation values for various values of m; n; and p.
12