I am trying to solve this problem where instead of uniform distribution, x and y are distributed EXP(1/b):

Suppose X and Y are two independent random variables, both uniformly distributed on (0,1). Let T1 = min(X; Y ) and T2 = max(X; Y ). 1) What is the joint cumulative distribution function and joint distribution of T1 and T2? 2) What is the conditional distribution of T2 given that T1 = t? 3) Are T1 and T2 independent?

For the joint cumulative distribution I use Bayes formula and I don't know how to proceed from there.

For the joint distribution function I would derivate the result according to each variable

For the conditional distribution I would rearrange Bayes formula from part 1 and derivate according to the variable corresponding to T1

As for independence, I can see that they are dependent but I need proof.

Any help (preferably a solution) would be greatly appreciated)


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