# Change of Variable Theorems

One Dimension :: Multiple Dimensions

## Changing variables: One dimension

## A simply invertible function

Let
, where
is a continuous r.v. and
is a
one-to-one, onto, measurable function. Then

So we can determine the distribution of . Let us now take a different point of view that will allow us to generalize to higher dimensions and develop and understand a commonly used formula.

Consider an interval along the
axis,

Suppose the function has a positive derivative. The interval along the axis, when is at the point . The probability that falls in its interval is the probability that falls in its interval:

where , or equivalently, . Then

That is

If we take the other case that
has a negative derivative, we
have to take
. Combining these together we
obtain

## Multiple inverses

It may happen that is not a uniquely invertible function. That is, for a given there may be more than one value of such that . For example, : then and are both inverses.

We will prove the concept for two solutions, Let
, assuming to be specific that the slope is positive at
and negative at
.

That is

From this,

This is sometimes written

In general, with solutions we have

## constant in an interval

If the function
is constant over any interval, then there is no
inverse, nor even multiple inverses. However, we can still compute
the distribution. Let
for
(i.e.,
constant). Then

Hence, there is probability mass at the point . This results in a c.d.f which is not continuous at that point.