Thread: 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.

Yes, you need to do a 2-2 transformation using jacobians
TYPE what you have so far and I'll look it over.

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.

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?

It is only two iid RVs, both exponentially distributed

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