Some applications of Dirac's delta function in Statistics for more than ...

Abstract
Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Vol. 3, Issue 1 (June 2008), pp. 42 – 54
(Previously, Vol. 3, No. 1)
42
Applications and Applied
Mathematics:
An International Journal
(AAM)
Some Applications of Dirac's Delta Function in Statistics for
More Than One Random Variable
Santanu Chakraborty
Department of Mathematics
University of Texas Pan American,
1201 West University Drive, Edinburg, Texas 78541, USA
schakraborty@utpa.edu
Received July 14, 2006; accepted December 7, 2007
In this paper, we discuss some interesting applications of Dirac's delta function in Statistics. We
have tried to extend some of the existing results to the more than one variable case. While doing
that, we particularly concentrate on the bivariate case.
Keywords: Dirac's Delta function, Random Variables, Distributions, Densities, Taylor's Series
Expansions, Moment generating functions.
1. Introduction
Cauchy, in 1816, was the first (and independently, Poisson in 1815) gave a derivation of the
Fourier integral theorem by means of an argument involving what we would now recognize as a
sampling operation of the type associated with the delta function. And there are similar examples
of the use of what are essentially delta functions by Kirchoff, Helmholtz and Heaviside. But
Dirac was the first to use the notationδ . The Dirac delta function (δ -function) was introduced by
Paul Dirac at the end of the 1920s in an effort to create the mathematical tools for the
development of quantum filed theory. He referred to it as an “improper function” in Dirac
(1930). Later, in 1947, Laurent Schwartz gave it a more rigorous mathematical definition as a
linear functional on the space of test functions D (the set of all real-valued infinitely
differentiable functions with compact support) such that for a given function f (x)
in D , the
value of the functional is given by the property (b) below. This is called the sifting or sampling
property of the delta function. Since the delta function is not really a function in the classical
sense, one should not consider the “value” of the delta function at x . Hence, the domain of the
delta function is D and its value, for f ∈ D and a given x 0 is ). ( 0 x f Khuri (2004) studied some
interesting applications of the delta function in statistics. He mainly studied univariate cases even
though he did give some interesting examples for the multivariate case. We shall study some
more applications in the multivariate scenario in this work. These might help future researchers

AAM: Intern, J., Vol. 3, Issue 1 (June 2008) [Previously, Vol. 3, No. 1] 43
in statistics to develop more ideas. In sections 2 and 3, we discuss derivatives of the delta
function in both univariate and multivariate case. Then, in section 4, we discuss some
applications of the delta function in probability and statistics. In section 5, we discuss
calculations of densities in both univariate and multivariate case using transformations of
variables. In section 6, we use vector notations for delta functions in the multidimensional case.
In section 7, we discuss very briefly the transformations of variables in the discrete case. Then,
in section 8, we discuss the moment generating function in the multivariate set up. We conclude
with few remarks in section 9.
2. Derivatives of the δ -function in the Univariate Case
In the univariate case, some basic properties satisfied by Dirac's delta function are:
where (x)
have,
∞
∫
−∞
(a) δ ( x)
dx = 1,
b
(b) ∫ f ( x)
δ ( x − x0
) dx = f ( x0
) for all a < x0
< b ,
a
f is any function continuous in a neighborhood of the point 0
∞
∫
−∞
f x)
δ ( x − x ) dx = f ( x ).
( 0
0
x . In particular, we
This is the sifting property that we mentioned in the previous section. If f (x)
is any function
with continuous derivatives up to the
th
x , then
b
∫
a
In particular, we have,
∞
∫
−∞
f
n order in some neighborhood of 0
( n)
n ( n)
f x)
δ ( x − x ) dx = ( −1)
f ( x ), n ≥ 0 for all a < x0
< b .
( 0
0
( n)
n ( n)
x)
δ ( x − x ) dx = ( −1)
f ( x ), n ≥ 0
( 0
0
(n)
for a given x 0 . Here, δ is the generalized
linear functional which assigns the value 1)
( )
f ( x )
n n
− to f (x)
.
th
n order derivative of δ . This derivative defines a
( 0
Now let us consider the Heaviside function H(x) unit step function defined by

44 Santanu Chakraborty
H ( x)
= 0 for x < 0
= 1 for x ≥ 1.
dH ( x)
The generalized derivative of H (x)
is δ (x)
, i.e., δ ( x)
= . As a result, we get a special
dx
th
case of the formula for the n order derivative mentioned above:
∞
∫
−∞
n ( i)
x δ ( x)
dx = 0 if i ≠ n
( 1)
n!
n
= − if i = n
3. Derivatives of the delta function in the bivariate case
Following Saichev and Woyczynski (1997), Khuri (2004) provided the extended definition of
delta function to the n -dimensional Euclidean space. But we shall mainly concentrate on the
bivariate case. As in the univariate case, we can write down similar properties for the bivariate
case as well. In the bivariate case, δ ( x, y)
= δ ( x)
δ ( y)
. So, if we assume f ( x,
y)
to be a
continuous function in some neighborhood of x , ) , then we can write
∫∫
ℜ×
ℜ
( 0 0 y
f x,
y)
δ ( x − x , y − y ) dxdy = f ( x , y ) ,
( 0 0
0 0
where ℜ is the real line.
Now, for this function f , if all its partial derivatives up to the th
n order are continuous in the
abovementioned neighborhood of x , ) , then,
( 0 0 y
∂ f ( x,
y)
δ …… (1)
n
( n)
n n
∫∫ f ( x,
y)
( x − x0
, y − y0
) dxdy = ( −1)
∑ Ck
|
k n−k
x=
x0
, y=
y0
ℜ×
ℜ
0≤k
≤n
∂x
∂y
n ( n)
where C k is the number of combinations of k out of n objects, δ ( x,
y)
is the generalized
th
n order derivative of δ ( x,
y)
. In the general p-dimensional case, by using induction on n , it
can be shown that
∞ ∞
∫... ∫
( n)
f ( x1,...,
x p ) δ ( x1
*
*
x1
,..., x p − x p ) dx1...
−∞
−∞
∂
= − ∑ ∑
∂ ∂
−
n n−k1
−...
−k
p 1
( n)
n
n!
f
( 1)
...
k1
k p
0 0 k1!...
k p!
x1
... x p
− dx p
| x = x
*
,

AAM: Intern, J., Vol. 3, Issue 1 (June 2008) [Previously, Vol. 3, No. 1] 45
where x = ( x1
,..., x ) ′ p ,
* *
= ( 1 ,..., ) ′ p x x
*
x and f is a function of p variables, namely,
x x , , x .
1, 2
p
4. Use of Delta Function to Obtain Discrete Probability Distributions
If X is a discrete random variable that assumes the values n a a 1,..., with probabilities n p p 1 ,...,
respectively such that ∑ pi
= 1, then, the probability mass function of X can be represented
1≤i≤
n
as p(
x)
= ∑ piδ
( x − ai
).
1≤i≤n
Now let us consider two discrete random variables X and Y which assume the values n a a 1 ,...,
and n b b 1 ,..., , respectively, and the joint probability P ( X = ai
, Y = b j ) is given by p ij for
i = 1,...,
m and j = 1,...,
n so that the joint probability mass function p ( x,
y)
is given by
p( x,
y)
p δ ( x − a ) δ ( y − b ) .
= ∑ ∑
1≤i≤m 1≤
j≤n
ij
i
Similarly, one can write down the joint probability distribution of any finite number of random
variables in terms delta functions as follows:
j
Suppose we have k random variables k X X 1 ,..., with X i taking values a ij , j = 1,
2,...,
ni
for
i = 1,
2,...,
k with probability k i i p 1 ... . Then, the joint probability mass function is
n
1
k
P( X = x1,...,
X k = xk
) = ∑ ∑ pi
( ) (
1
i δ x k 1 − a1i
δ
x
1
k − a
n
1 kik
i1
= 1 ik
= 1
As an example, we may consider the situation of multinomial distributions. Let
X X ,... X
n p , p ,..., p . Then,
1, 2 k follow multinomial distribution with parameters , 1 2 k
n!
i1
P( X 1 = i1,
,
X k = ik
) = p1
p
i ! i
!
1
k
ik
k
where i 1, , ik
add up to n and k p p , 1 add up to 1. In terms of delta function, the joint
probability mass function is
PX ( 1 = x1, X2 = x2,..., Xk = xk)
= ...
n!
i1 i2
ik
p1 p2 ... pk δ( x1−i1) δ( x2 −i2)... δ( xk −ik)
i ! i !... i !
∑∑ ∑ .
i1 i2 ik1 2 k
)

46 Santanu Chakraborty
We can also consider conditional probabilities and think of expressing them in terms of the δ -
function. Let us go back to the example of the two discrete random variables X and Y , where
X takes the values m a a a , , , 1 2 and Y takes the values n b b b , , , 1 2 . Then, the conditional
probability of Y = y given X = x is given by
PY ( = y, X= x)
p( y| x) = PY ( = y| X= x)
=
PX ( = x)
∑∑
pijδ( x−ai) δ( y−bj) pxy ( , ) 1≤≤ i m1≤j≤n = =
.
px ( ) pδ( x−a) ∑
1≤≤
i n
i i
5. Densities of Transformations of Random Variables Using δ -function
If X is a continuous random variable with a density function f (x)
and if Y = g(X
) is a
function of X , then the density function of Y , namely, h (y)
is given by
∞
∫
−∞
h ( y)
= f ( x)
δ ( y − g(
x))
dx .
We can extend this to the two-dimensional case. If X and Y are two continuous random
variables with joint density function f ( x,
y)
and if Z = φ1(
X , Y ) and W = φ2
( X , Y ) are two
random variables obtained as transformations from ( X , Y ) , then the bivariate density function
for Z and W is given by
∞ ∞
( z,
w)
∫ ∫ f ( x,
y)
−∞−∞
( z −φ1
( x,
y))
δ ( w −φ
2
h = δ ( x,
y))
dxdy ,
where z and w are the variables corresponding to the transformations φ1( X , Y ) and φ 2 ( X , Y ) .
This has obvious extension to the general p-dimensional case.
Khuri (2004) gave an example of two independent Gamma random variables X and Y so that
X and Y are gamma random variables with distributions Γ ( λ,
α1)
and Γ ( λ,
α 2 ) respectively. If
we denote the densities as f 1 and f 2 respectively, then we have,
α1
λ α1−1
−λx
f1(
x)
= x e , for x > 0
Γ(
α )
1
= 0 , for x ≤ 0 .

AAM: Intern, J., Vol. 3, Issue 1 (June 2008) [Previously, Vol. 3, No. 1] 47
f
2
α 2 λ α 2 −1
−λy
( y)
= y e , for y > 0
Γ(
α )
2
= 0 , for y ≤ 0 .
X
In that case, if we define Z = and W = X + Y , then, Z is distributed as Beta with
X + Y
parameters α 1 , α 2 and W is distributed as Gamma with parameter values 1 and α 1 + α 2 . From
now on, we shall use β ( a,
b)
to indicate a Beta random variable with positive parameters a , b
and Γ ( c,
k)
to indicate a Gamma distribution with positive parameters c and k .
Now let us consider 3 random variables X 1 , X 2 , X 3 distributed independently so that X i is
1
distributed as gamma, i.e., Γ ( , α i ) for i = 1, 2, 3, and if we define
2
X 1
Y1
=
X 1 + X 2
,
X 1 + X 2
Y2
=
X 1 + X 2 + X 3
,
Y = X + X + X .
3
1
2
Then, we have,
Y 1 distributed as β ( α1,
α 2 ) ,
Y 2 distributed as β ( α1
+ α 2 , α 3)
,
1
Y 3 distributed as Γ ( , α 1 + α 2 + α 3)
.
2
3
This can be shown following exactly the same technique used in Khuri (2004) which is a one
step generalization of the result proved by him. So, we define
*
x1
x1
+ x2
δ ( x1, x2
, x3,
y1,
y2
, y3
) = δ ( − y1)
δ (
− y2
) δ ( x1
+ x2
+ x3
− y3
) .
x + x x + x + x
The joint density of Y1, Y2 and Y3 is given by
1
1 2 3
∞ ∞ ∞
∫∫∫
0 0 0
1 2 3 1 2 3 1 2 3 1
∞ ∞ ∞ x1
+ x2
+ x3
1
α1−1
α 2 −1
α3
−1
−
2
x1
x2
x3
e
α1+ α 2 + α3
Γ(
α1)
Γ(
α 2 ) Γ(
α 3 ) ∫∫∫
0 0 0
*
g( y , y , y ) = f ( x , x , x ) δ ( x , x , x , y , y , y ) dx dx dx
=
2
Using properties of the delta function,
2
1
2
3
2
1
2
*
δ ( x , x , x , y , y , y ) dx dx dx
2
3
1
2
3
1
2
3

48 Santanu Chakraborty
the innermost integral (integral with respect to x 1)
∞ x1
+ x2
+ x3
α1−1
−
2 *
= ∫ x1
e δ ( x1,
x2
, x3,
y1,
y2
, y3
) dx1
0
= ∫
∞ x1
+ x2
+ x3
α1−1
− x
2
1
x1
+ x2
x1
e δ ( − y1)
δ (
− y2
) δ ( x1
− ( y3
− x2
− x3
)) dx1
x
0
1 + x2
x1
+ x2
+ x3
y
3
−
α1−1
y3
− x2
− x
2
3 y3
− x3
= ( y3
− x2
− x3
) e δ (
− y1)
δ ( − y2
) .
y − x
y
3
Next, we use delta function properties to deal with the second integral (the integral with respect
to x 2 ) as
∞
y3
α −
−
− − −
2 1
α 1 y3
x
2
2 x
1
3
∫ x2
( y3
− x2
− x3
) e δ (
− y1)
dx2
y − x
0
3 3
∞
α1−1
y3
α 1 ( 3 2 3)
2 − y − x − x −
2
∫ x2
e ( x2
− ( y3
− x3
)( 1−
y1))
dx2
y
0
3 − x3
y3
1
−
α1
+ α 2 −1
α1−
α 2 −1
2
( y3
− x3
) y1
( 1−
y1)
e
= δ
=
Then, we use delta function properties for the outermost integral (without the constant terms) is
∞
∫
0
x
y3
α −1
−
−
+ −1
1
−1
1
3 α1
α 2 α1
α 2 2
3 ( y3
− x3
) y1
( 1−
y1)
e δ ( x3
− y3
( 1−
y2
)) dx3
y3
y3
α1
−1
1
1
1
1
−
α 2 − α1+
α 2 −
α3
− α1+
α 2 + α3
− 2
= y1
( 1−
y1)
y2
( 1−
y2
) y3
e
Finally, putting the constant terms together, we get
g(
y1,
y2
, y3
) =
2
1+
α 2 + α3
This completes the proof.
α
1
y
Γ(
α ) Γ(
α ) Γ(
α )
1
2
3
3
α1−1
1
( 1−
y
α 2 −1
α1+
α 2 −1
1)
y2
6. Vector notations for delta functions in the multidimensional case
3
( 1−
y
y3
1
1
−
α3
− α1+
α 2 + α3
− 2
2 ) y3
e
.

AAM: Intern, J., Vol. 3, Issue 1 (June 2008) [Previously, Vol. 3, No. 1] 49
In the multidimensional case, if the transformation is linear, i.e., Y = AX where Y and X are
m × 1 and n × 1 vectors respectively and A is an m × n matrix, then we can express g (y)
, the
density of Y , in vector notation in terms of f (x)
, the density of X as follows
where
So,
∞
g ∫ ∫
( y) = ... f ( x)
δ ( y − Ax)
dx
, (2)
−∞
∞
−∞
δ y ) = δ ( y )... δ ( y ) .
( 1 m
δ ( y − Ax)
= δ ( y1 − a1
x)...
δ ( ym
− am
x)
,
T
1
T
m
T
T
where a ,..., a are the rows of the matrix A . Now, in the one-dimensional set up, if a is a
δ ( x)
scalar, then δ (ax)
is given by δ ( ax)
= . Similarly, in the multidimensional set up, if
| a |
Y = AX as above and A is a nonsingular matrix so that m = n , then, we must have
−1
δ ( x − A y)
δ ( y − Ax)
=
. (3)
| A |
This is because of the following: since the transformation is nonsingular, we have
1 −1
g ( y)
= f ( A y)
| A |
and therefore, from (2)
∞
A y x A y Ax x
1 −
( ) = ... f ( ) | | δ ( − ) d .
f ∫ ∫
But we know that
∞
−∞
∞
∫ ∫
−∞
∞
−∞
−1
−1
... f ( x)
δ ( x − A y)
dx
= f ( A y)
.
−∞
Therefore, (3) follows. Similarly, if Y = AX + b where A is a nonsingular matrix and b is a
vector, then, we have

50 Santanu Chakraborty
−1
δ ( x − A (y − b))
δ ( y − (AX + b))
=
.
| A |
Using this, one can conclude that if X is multivariate normal with μ as the mean vector and ∑
as the covariance matrix, then, for a nonsingular transformation A and a constant vector b , the
transformed vector Y = AX + b follows multivariate normal with Aμ + b as the mean vector
T
and A ∑ A as the variance-covariance matrix.
7. Transformation of Variables in the Discrete Case
Transformation of variables can be applied to the discrete case as well. If X is a discrete random
variable taking the values n a a , , 1 with probabilities n p p , , 1 and if ) (X g Y = is a
transformed variable, then the corresponding probability mass function for Y is given by
∞
∫ p(
x)
( y − g(
x))
dx = ∑
q(
y)
= δ p δ ( y − g(
a )) .
−∞
n
1
In the two-dimensional case, if p ( x,
y)
is the joint probability mass function for the two
variables X and Y , then q ( z,
w)
, the joint probability mass function for the transformed pair
Z = φ ( X , Y ) and W = φ ( X , Y ) is given by
1
2
∞ ∞
( z,
w)
∫ ∫ p(
x,
y)
δ ( z −φ1
( x,
y))
δ ( w −φ
2
−∞−∞
m n
q = ( x,
y))
dxdy
=
∑∑
1 1
p δ ( z −φ
( a , b )) δ ( w −φ
( a , b ))
ij
1
i
j
Where the pair ( X , Y ) is discrete having the values ( a i , b j ) with probabilities p ij for i = 1,
2,...,
n
and j = 1,
2,...,
m .
8. Moments and Moment Generating Functions
In the univariate set up, the
∞
k
k
∫ x p x)
dx = ∫ x ∑
−∞
∞
−∞
th
k non-central moment of X is written as
1≤i≤
n
∑
2
1≤i≤
n
i
i
∞
j
i
k
∫ x ( x − ai
) dx = ∑
( p δ ( x − a ) dx = p δ
p a .
i
i
i
−∞
1≤i≤
n
In the bivariate set up, the non-central moment of order (k,l) for (X,Y) is given by
∞ ∞
∫ ∫
−∞−∞
x
k
∞ ∞
k l
∫ ∫ x y ∑∑
l
y p(
x,
y)
dxdy = p δ
( x − a ) δ ( y − b ) dxdy
−∞−∞
m n
1 1
ij
i
j
i
k
i

AAM: Intern, J., Vol. 3, Issue 1 (June 2008) [Previously, Vol. 3, No. 1] 51
=
m
n
∞ ∞
k l
∑∑ pij
∫ ∫ x y ( x − ai
, y − b j ) dxdy = ∑∑
i=
1 j=
1 −∞−
∞
i=
1 j=
1
For example, if ( X , Y ) follow trinomial distribution, then
n n−k
n!
k l
p(
x,
y)
= ∑∑
p1
p2
( 1−
p1
− p2
)
k!
l!
( n − k − l)!
k = 0 l=
0
m
n
k
i
δ a b p .
n−k
−l
δ ( x − k)
δ ( y − l)
.
As a result, the corresponding non-central moment of order ( r , s)
is given by
∞ ∞
r s
∫ ∫ x y p x,
y)
dxdy = ∑∑
−∞−
∞
k = 0 l=
0
n−k
−l
n n−k
n!
k l
r s
( p1
p2
( 1−
p1
− p2
) ak
bl
.
k!
l!
( n − k − l)!
The most interesting part in Khuri's article is the representation of the density function of a
continuous random variable X in terms of its non-central moments. Thus, if f (x)
is the density
function for X , then f (x)
is represented as
f ( x)
= ∑
0≤m<
∞
( −1)
m!
m
µ δ
m
( m)
( x)
,
( m)
th
th
where δ ( x)
is the generalized m derivative of δ (x)
and µ m is the m order non-central
moment for the random variable X . One can see a proof of this result in Kanwal (1998). Let us
briefly mention the technique used by him to derive the above expression. He showed that, for
any real function ψ defined and differentiable of all orders in a neighborhood of zero,
n
( −1)
( n)
f , ψ 〉 = 〈
µ δ ( x),
ψ 〉
0≤
<
n ,
n n!
〈 ∑ ∞
where 〈 f , ψ 〉 , the inner product between the functions f and ψ , is defined as
∞
∫
−∞
〈 f , ψ 〉 = f ( x)
ψ ( x)
dx .
This leads us to conclude
n
( −1)
( n)
f( x) = ∑ µδ ( )....
0 n
n x
(4)
≤

52 Santanu Chakraborty
Then,
∞
∫
−∞
( n)
n
( n)
n ( n)
ψ ( 0)
= ( −1)
ψ ( x) δ ( x)
dx = ( −1)
〈 ψ , δ 〉 .
These two steps give us the relation (4).
When we move to the two-dimensional scenario, we shall have to use Taylor's series expansion
for a two-dimensional analytic function ψ about the point (0,0) which is given by
1 ∂ ∂ n
ψ ( x,
y)
= ∑ [ x + y ] ψ ( x,
y)
|
0≤
n<
∞ n!
∂x
∂y
1
= ∑
∂
x=
0,
y=
0
∞ ∞
n
n
k n−k
∑ Ck
ψ ( x,
y)
|
k n k
x=
y=
x y
−
0,
0
(5)
n=
0 n!
k = 0 ∂x
∂y
Now, here also, we follow the same technique and so we compute 〈 f , ψ 〉 which is defined as
∞ ∞
∫ ∫
〈 f , ψ 〉 = f ( x,
y)
ψ ( x,
y)
dxdy
−∞−∞
Now using Taylor’s series expansion of ψ ( x,
y)
about ( 0 , 0 ) y x from (5) and assuming that
interchange of integrals with summations permissible, we get
∞ ∞
1
∂
〈 f , ψ 〉 = ∫ ∫ f ( x,
y)[
∑ ≤n<
∞ ∑
ψ ( x,
y)
|
0 n!
=
=
−∞−∞
Now we define the
∞ ∞
n
1 ∂
k!
( n − k)!
∂x
∂y
n
n
C
0≤k
≤n
k k n−k
∂x
∂y
∑ ∑ ψ ( x,
y)
|
0≤n< ∞ 0≤k ≤n
k n−k
x=
0 , y=
0 ∫ ∫
∑ ∑
0≤n<
∞
th
(,)
0≤k
≤n
1
k!
( n − k)!
∂
k
∂x
∂y
x=
0,
y=
0
∞ ∞
−∞−∞
x
k
y
f ( x,
y)
x
n−k
n
k n−k
ψ ( x,
y)
|
−
x=
y=
〈 f x y 〉
n k
0,
0 , .
rs order non-central moment for the pair ( X , Y ) as
r s
r s
µ f ( x,
y)
x y dxdy = 〈 f , x y 〉 .
r,
s
= ∫ ∫
−∞−∞
Then from above,
n
1 ∂
〈 f , ψ 〉 = ∑ ≤ < ∞∑
ψ ( x,
y)
|
0 n 0 ≤k
≤n
k n−k
x=
0,
y=
0 µ k , n−k
k!
( n − k)!
∂x
∂y
1
n ( k ) ( n−k
)
= 〈 ∑ ∑
µ −
〉
≤ < ∞ ≤ ≤
− ( 1)
δ ( ) δ ( ) , ψ
0 n 0 k n
k , n k x y .
k!
( n − k)!
k
y
] dxdy
n−k
dxdy

AAM: Intern, J., Vol. 3, Issue 1 (June 2008) [Previously, Vol. 3, No. 1] 53
The last equality follows because of the following relation
n
( k ) ( n−k
)
n ∂
〈 ψ , δ ( x) δ ( y)
〉 = ( −1)
ψ ( x,
y)
|
k n−k
x=
0,
y=
0 .
∂x
∂y
Therefore, we have
1
f ( x,
y)
= ∑ ≤ n< ∞∑
δ
0 0≤k
< n k!
( n − k)!
n ( k ) ( n−k
)
µ k , n−k
( −1)
δ ( x)
( y)
.
Now, the non-central moment of order ( r , s)
is given by
∞ ∞
∫ ∫
−∞−∞
But, we also have
∫∫
∞ ∞
r s
r s
f ( x,
y)
x y dxdy = ∫ ∫ x y ∑ ∑
−∞−∞
0≤n< ∞ 0≤k
≤n
r s ( i)
( j)
x y δ ( x)
δ ( y)
dxdy = 0,
if i ≠ r or j ≠ s
1
k!
( n − k)!
r+ s
= ( −1)
r!
s!
, if i = r and j = s .
Therefore, the non-central moment of order ( r , s)
reduces to µ r, s .
n ( k ) ( n−k
)
µ k , n−k
( −1)
δ ( x)
δ ( y)
dxdy .
When we talk about the moment generating function in the one variable case, we have
∞
n
tX tx ( −1)
(
φ ( t)
= E(
e ) = ∫ e ∑ µ
0≤n<
∞
nδ
n!
=
=
=
−∞
n
( −1)
n!
( −1)
n!
µ n
t
n
∞
∑ µ
0≤n<
∞
n ∫
∑0≤ n<
∞
∑0 ≤n<
∞ !
n
n
−∞
e
tx
µ ( −1)
.
n
δ
n
( n)
d
dx
( x)
dx
n
n
e
tx
|
n)
x=
0
( x)
dx
In the two-variable case, the moment generating function is given by
φ ( s,
t)
∞ ∞
sX + tY
sx+
ty
= E(
e ) = ∫ ∫ e f ( x,
y)
dxdy
−∞−∞
∞ ∞
sx+
ty
1
= ∫ ∫ e ∑0≤ n< ∞∑
0≤k
< n k!
( n − k
−∞−∞
=
1
k!
( n − k)!
µ
)!
k , n−k
∞ ∞
n
∑ ∑ µ
≤ < ∞ ≤ <
− −
0 n 0 k n
k , n k ( 1)
∫ ∫
−∞−∞
n
( −1)
δ
e
sx+
ty
δ
( k )
( k )
( x)
δ
( x)
δ
( n−k
)
( n−k
)
( y)
dxdy
( y)
dxdy

54 Santanu Chakraborty
=
=
∑ ∑
1
µ
k!
( n − k)!
0≤ n< ∞ 0 ≤k
< n
k , n−k
∑ ∑
0≤ n< ∞ 0 ≤k
< n
k , n−k
s
k n−k
s t
µ .
k!
( n − k)!
t
k n−k
In the general p -dimensional case, the moment generating function could be obtained exactly in
the similar fashion so that, it is given by
k
k
1 k2
p
t t
1 t2
p
φ( t 1, t2,
, t p ) = ∑0 n ∑
0 k n ∑
µ
0 k
k k k
p n k k
, , ,
1 1 p 1
1 2 .
≤ < ∞ ≤ < ≤ < − − − −
p
k ! k ! k !
9. Concluding Remarks
The study of generalized functions is now widely used in applied mathematics and engineering
sciences. The δ -function approach provides us with a unified approach in treating discrete and
continuous distributions. This approach has the potential to facilitate new ways of examining
some classical concepts in mathematical statistics. However, some interesting applications can
be found in the paper by Pazman and Pronzato (1996). In this paper, the authors use delta
function approach for densities of nonlinear statistics and for marginal densities in nonlinear
regression. We are also looking forward to obtain some interesting applications of the delta
function in statistics.
Acknowledgement:
I am sincerely thankful to Professor Lokenath Debnath in the University of Texas-Pan American
for bringing this problem to my notice.
REFERENCES
Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, Oxford University Press.
Hoskins, R.F. (1998). Generalized Functions, Ellis Horwood Limited, Chichester, Sussex,
England.
Kanwal, R.P. (1998). Function Theory and Technique (2nd Edition), Boston, MA, Birkhauser.
Khuri, A.I. (2004). Applications of Dirac's delta function in statistics, International Journal of
Mathematical Education in Science and Technology, 35, no. 2, 185-195.
Pazman, A and Pronzato, L. (1996). A Dirac-function method for densities of nonlinear statistics
and for marginal densities in nonlinear regression, Statistics & Probability Letters, 26, 159-
167.
Saichev, A.I. and Woyczynski, W.A. (1997). Distributions in the Physical and Engineering
Sciences, Boston, MA, Birkhauser.
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2
p