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

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
*
,