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

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 .