I cant comment on your solution but i think there might be an easier way
Spoiler:
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!
Are you sure of the second equality ?
Anyway, if you really want to do something along the lines of what you've already done, can't you write it this way :
But
So finally,
Did it go wrong somewhere ? oO
Springfan25 : there's a problem with your boundaries, since for example appears after having integrated with respect to , same for .
Are you sure about that one?
it must be true that
it must also be true that
You propose
(2) and (3) Together imply:
But i think (4) contradicts (1.1)
I can only see 2 ways that (4) could be consistent with (1.1)
either (not true in general)
or . This is clearly not the case for any given value of x3 (eg, consider x3=0), and therefore (i think?!) not true....but im not sure about that reasoning.
In the unlikely event that i am right about the above, i think the integration approach could still work. I may have the limits wrong but there is a constructable region R (albeit unbounded) on which the integral needs to be done. Here's my second go:
Spoiler: