Change of Variables

Ponder...

Change of Variables

For strictly monotone with differentiable inverse, has density .

Theorem 4.16

Let have density , and let be a function which is strictly monotone and is differentiable. Then has density

Proof of the monotone case

Suppose first that is strictly increasing. Then

and differentiating gives the density , which is non-negative because .

If is strictly decreasing, then

and differentiating gives the density , non-negative because . Both cases combine into .

Theorem 4.28

Let be a random variable with values in and density . Let be a bijection with a continuous derivative on , and

where is the Jacobian determinant of .

Let be independent and set , . The map has Jacobian determinant

so the density of is

for and . The joint density factorises into a function of times a function of , so and has density for , independently of each other.