Ratio of uniforms method proof

[crazy2006]

Hi everybody:
I am trying to solve this question:
[URL=http://imageshugger.com/][PLAIN]http://imageshugger.com/images/wviidazp3mf06ej32.jpg[/URL]

I have made the following:
u=y
v=xy

Calculated the Jacobian, which is |-y|=y
then as I know that is unnormalized I need to normalized first using a double integral as a normalizing factor, the thing is that I am stuck there
please could you help me to get the joint pdf of X and Y
thanks

 

[EnumaElish]

Homework?

 

[crazy2006]

no homework, this question is posted in a book about probability and also it is somewhat explained into the paper of 1977 of Kinderman and Monahan, but as I said I got stucked into the joint pdf and not get the answer
any help would be valuable

 

[crazy2006]

nevermind I got the answer, thanks anyway

 

[blue_raver22]

Hello,

Thank you for sharing your progress so far. The Ratio of Uniforms method is a technique used for generating random numbers from any continuous probability distribution. It involves using a uniform random number generator and transforming it into a sample from the desired distribution.

In order to find the joint probability density function (pdf) of X and Y, you will need to use the Jacobian transformation and the normalizing factor you mentioned. The joint pdf can be calculated using the formula:

f(x,y) = f(u,v) * |J(u,v)| / K

Where f(u,v) is the pdf of the transformed variables u and v, |J(u,v)| is the absolute value of the Jacobian determinant, and K is the normalizing factor.

In this case, since u = y and v = xy, the Jacobian determinant is simply |J(u,v)| = |y| = y. And the normalizing factor can be found by integrating the joint pdf over the entire range of x and y, which will give you a value of 1.

Therefore, the joint pdf of X and Y can be written as:

f(x,y) = f(u,v) * y

You can then substitute in the pdf of u and v, which in this case is the uniform distribution. So if u and v are both between 0 and 1, then f(u,v) = 1. This gives you:

f(x,y) = y

Which is the joint pdf of X and Y. I hope this helps you to continue with your problem. Good luck with your research!