Let {N(t): t≥0} be a Poisson process of rate 1 and let S_r be the time to the rth point. Calculate the density function of X = S_2/S_6 using the Jacobian transformation method.

To use the Jacobian method, I need to define another random variable Y that is a function of S_2 and S_6.

X=S_2/S_6

Y=???

I know that S_2~gamma(2,1) and S_6~gamma(6,1), but I am having some trouble defining the other random variable Y, and the problem is thatS_2 and S_6 are not independent.

The Jacobian method expresses the joint density of X and Y in terms of S_2 and S_6, so I must first obtain the joint density of S_2 and S_6. But now it seems like that I have no way of obtaining the joint density of S_2 and S_6 because they are NOT independent random variables (S_2<S_6 always).

So how can we solve this problem using the Jacobian method?

Any help is much appreciated!