This is what I need to do but don't know where to start, please give me some pointers. It has to do with expected value of nonnegative random variable.
Let
Show that
I know that ,
but what is and ?
Where do i even start?
Hello,
I don't understand.. You're given this problem, but you don't even know what p(k) and are ?
That's not normal !!
Anyway, in general, is the probability distribution, and is the cumulative distribution function.
Hence
Since ,
Can you see a pattern when you sum these ?
Now it looks logical that
But I can't manage to find a way to prove it closely. Maybe I'll find tomorrow, or someone else will
-----------------------------------------------------
You can prove by induction, for , that :
Then take n to infinity... But I don't know how to deal with it :s
If , then the right-hand side goes to infinity (it is greater than ), so that the equality holds.
Suppose now . In order to take the limit as , we have to justify .
This can be written . However, we have , and the bounded convergence theorem shows that the right-hand sides converges to 0 since .
-------------
There's another solution without taking a limit. This involves Fubini theorem for double series.
Indeed, if you write
,the equality amounts to interchange the sum signs, I let you get convinced of this.
Thanks for all the help all. I know this might sound stupid, it's the first time that I did any kind of stats, but I thought Fubini's Theorem is used on integrals in vector calculus. How did you turn that into the 2 sums? I understand the reasoning behind what moo did the first time the most. Am I wrong to think so?
What Moo did is also "Fubini" (interchanging sums) but for a finite sum; in this case we don't say Fubini because it is always true by simple properties of the sum.
Let me repeat what Moo wrote in a pretty way:
If you sum line after line (there are infinitely many), you get .
If you sum column after column, you get .
In order to justify why these sums are the same, you need a theorem. Fubini theorem holds for double integrals indeed, but there is a very similar theorem for double sums, and I think it is customary to call it Fubini as well.
Anyway, this theorem says that if we have nonnegative numbers (like in this case), then we can interchange the sum signs (whether the sum is finite or not):
.
This amounts to adding the columns or the lines in the previous sketch.
NB: The theorem also gives a condition for doing the same in case there are negative numbers:
if , then .
--
You may prefer not to use this theorem about double series and sum only lines of the previous sketch to get the equality Moo gives:
Then you need to take the limit as tends to . I gave a possible justification of the limits in the first part of my previous post.
Hmmm ... I was vainly trying to deal with Fubini, but there was something wrong... :
Isn't the red k an i ?
Thereafter, it gives... very easily the solution... That's great ! Thinking of the indicator function was brilliant lol.
Thanks for that Fubini stuff, Laurent.