This has been bothering me for a while now:

Q: If are independent exponential random variables with rates , find

(a)

Here's what I figured out so far:

Let Then is and is independent of . Now observe that . We now see that

Now, take note that the distribution of and given that they are larger than is the same as that of and , so

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At this stage, I'm not sure how to tie it all together to get . Is it valid for me to state that we should simply get ?

I would appreciate any help with this!