Joint PDF of two independent exponential RV

• Nov 30th 2009, 07:20 PM
StatRookie
Joint PDF of two independent exponential RV
We know X and Y are iid distributed e^-x.

I am having trouble finding the joint pdf of Min(X,Y)/Max(X,Y).
I have already found the pdf of the min(x,y) and Max(x,y), I know that I cannot use convolutions or assume that the min(x,y) and Max(x,y) are independent. Would I need to find the Jacobian of Min(x,y) , Max(x,y) vs. X and y or is there some other method I am missing.
• Nov 30th 2009, 09:20 PM
matheagle
Yes, you need to do a 2-2 transformation using jacobians
TYPE what you have so far and I'll look it over.
• Nov 30th 2009, 09:39 PM
StatRookie
Wow that was fast. Thank-you in advance.

I let z=min(x,y) and found f(z)= 2exp(-2z)

and q= max(x,y) and found f(q)= 2exp(-q)-2exp(-2q).

I want to use the transformation method so that I may find the values of the min and max in terms of x and y, then substitute into the exponential RVs which I know are independent. I would also multiply by the Jacobian. I understand all of these things; but I am unable to setup the jacobian itself. I previously was able to evaluate the ratio of two independents, but I know that these order statistics are dependent and that also is causing me trouble.
• Nov 30th 2009, 10:00 PM
matheagle
NO, you need the joint distribution of the min and max.
THEY are not independent.
And is it a sample of just 2 or more?
• Dec 1st 2009, 05:45 AM
StatRookie
It is only two iid RVs, both exponentially distributed
• Dec 1st 2009, 07:38 AM
matheagle
1 Get the joint density of min and max of two iid exponentials.
Note that the region is min<max, they are not independent.

2 Tranform to W=min/max and say Z=min (Z=max might be better, I need to work thru to see). Here the jacobian is easy. Obtain the region too

3 Integrate out the Z