Poisson Process - proving independence

[violetvanny]

Hello,
I've just started learning about the poisson process, and I was hoping I could get some help with the following problem:

Assume that S and T are independent exponentially distributed. Set U=min(S,T) and V=max(S,T). Prove that U and W = V - U are independent.

Thanks!

 

[Romsek]

Can you come with the distribution for U?

Then can you come up with W in terms of S and T?

 

[violetvanny]

Can you come with the distribution for U?

Then can you come up with W in terms of S and T?

Sorry I'm really lost on this problem - all I have so far is this:

P(U>a) = P(min(S,T)>a) = P(S>a,T>a) = P(S>a)P(T>a) = e^-a(λ1+λ2)
for some a>0

but I'm not sure how to connect this to independence, also I'm not sure how to start with W.