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

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
)