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Transformation of variables in joint distributions

Transformation of variables in joint distributions

Suppose the random variables X i , i = 1, 2, . . . , n, are described by the multivariate PDF f(x 1 ,
x 2 , . . . , x n ). If we wish to consider random variables Y j , j = 1, 2, . . . , m, related to the X i by
Y j = Y j (X 1 , X 2 , . . . , X m ) then we may compute g(y 1 , y 2 , . . . , y m ), the PDF for the Y j , in a similar
way to that in the univariate case by demanding that

|f(x 1 , x 2 , . . . , x n )dx 1 dx 2 . . . dx n | = |g(y 1 , y 2 , . . . , y m )dy 1 dy 2 . . . dy m |.

From the discussion of changing the variables in multiple integrals given it follows that, in the
special case where n = m,

g(y 1 , y 2 , . . . , y m ) = f(x 1 , x 2 , . . . , x n )|J|,

where

∂x 1 ∂x n
. . .
∂(x 1 , x 2 , . . . , x n ) ∂y 1 . ∂y 1
.
. ,
J ≡ = . . . . . .
∂(y 1 , y 2 , . . . , y n )
∂x 1 ∂x n
. . .
∂y n ∂y n
is the Jacobian of the x i with respect to the y j .
Convolutions

Asaparticularconsequenceoftheaboveformulawecanshowthatthedensityfunctionofthesum
of two continuous random variables X and Y , i. e. of V = X + Y, having joint density function
f(x, y) is given by
∫
x
g(u) = f(x, u − x)dx (7.13)
−∞
In the special case where X and Y are independent, f(x, y) = f X (x)f Y (y) and (7.13) reduces to

∫
+∞
g(u) = f X (x)f Y (u − x)dx
−∞
which is called the convolution of f X and f Y , abbreviated f X ∗ f Y . The following are some
important properties of the convolution:

1. f 1 ∗ f 2 = f 2 ∗ f 1 ;

2. f 1 ∗ (f 2 ∗ f 3 ) = (f 1 ∗ f 2 ) ∗ f 3 ;

3. f 1 ∗ (f 2 + f 3 ) = f 1 ∗ f 2 + f 1 ∗ f 3 .

These results show that f 1 , f 2 , f 3 obey the commutative, associative and distributive laws of
algebra with respect to the operation of convolution.

Important joint distributions

In this section we will examine two important multivariate distributions, the multinomial
distribution, which is an extension of the binomial distribution, and the multivariate Gaussian
distribution.

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