# Change of Variable Theorems

One Dimension :: Multiple Dimensions

## Changing Variables: Multiple dimensions

Consider now multiple variables. Let
where

We go with the equal probability idea. The probability of falling in a region in space should be the same as the probability of falling in the corresponding region in space. We'll draw pictures in two dimensions, but the concepts apply to higher dimensions.

Suppose the region maps to the region in the coordinates. Equating probabilities We have

We need to evaluate .

The region
is a parallelepiped described by the vectors

Fact: Recall from calculus that that (signed) area of the parallelepiped described by the vectors and is obtained from the cross product. Let us express this in matrix form

In our case, we have

The (signed) area is then

Let

This matrix of partial derivatives is called the

**Jacobian**of the function .

Back to probabilities. We have

or

or

or, in general for an invertible function ,

## Many-to-one mappings

Let
where
and
are jointly distributed
random variables. For a given value of
, the inverse may form a
curve in
space. Let
denote the region in the
plane such that
. (This may not be a
connected region.) Then

so

Let denote the region of the plane such that . then

Sometimes it is helpful to introduce an auxiliary variable, then integrate it out.

Suppose
. Introduce the auxiliary variable
. Then the
inverse functions are straightforward to compute,

The Jacobian is

The joint density is

Then the density of can be obtained by integration: