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$begingroup$

I'm reading a book but I probably don't remember all the math needed to solve this.
The book says that:

x, y are two random variables such that y=g(x) where g is an invertible, continuous and differentiable transformation.

We have this property:
$|p_y(g(x))dy| = |p_x(x)dx|$ (1)

And solving from the equation you obtain:
$p_y(y)=p_x(g^-1(y))left|dfracpartial xpartial y right| $ (2)

or equivalently

$p_x(x)=p_y(g(x))left|dfracpartial g(x)partial x right|$ (3)

Passing from (1) to (2) I'm not sure how to deal with the differentials.

And passing from (2) to (3) i'm almost completely lost.

Can you please help me?

probability

share|cite|improve this question

asked Mar 21 at 12:49

Diego MoyaDiego Moya

11

$endgroup$

add a comment | 

0

$begingroup$

I'm reading a book but I probably don't remember all the math needed to solve this.
The book says that:

x, y are two random variables such that y=g(x) where g is an invertible, continuous and differentiable transformation.

We have this property:
$|p_y(g(x))dy| = |p_x(x)dx|$ (1)

And solving from the equation you obtain:
$p_y(y)=p_x(g^-1(y))left|dfracpartial xpartial y right| $ (2)

or equivalently

$p_x(x)=p_y(g(x))left|dfracpartial g(x)partial x right|$ (3)

Passing from (1) to (2) I'm not sure how to deal with the differentials.

And passing from (2) to (3) i'm almost completely lost.

Can you please help me?

probability

share|cite|improve this question

asked Mar 21 at 12:49

Diego MoyaDiego Moya

11

$endgroup$

add a comment | 

$begingroup$

I'm reading a book but I probably don't remember all the math needed to solve this.
The book says that:

x, y are two random variables such that y=g(x) where g is an invertible, continuous and differentiable transformation.

We have this property:
$|p_y(g(x))dy| = |p_x(x)dx|$ (1)

And solving from the equation you obtain:
$p_y(y)=p_x(g^-1(y))left|dfracpartial xpartial y right| $ (2)

or equivalently

$p_x(x)=p_y(g(x))left|dfracpartial g(x)partial x right|$ (3)

Passing from (1) to (2) I'm not sure how to deal with the differentials.

And passing from (2) to (3) i'm almost completely lost.

Can you please help me?

probability

share|cite|improve this question

asked Mar 21 at 12:49

Diego MoyaDiego Moya

11

$endgroup$

share|cite|improve this question

asked Mar 21 at 12:49

Diego MoyaDiego Moya

11

share|cite|improve this question

asked Mar 21 at 12:49

Diego MoyaDiego Moya

11

asked Mar 21 at 12:49

Diego MoyaDiego Moya

11

asked Mar 21 at 12:49

Diego MoyaDiego Moya

11