This is a quick one, if you guys can help...
I'd like to make sure that the Jensen inequality comes out with the equal sign for the following problem (i is just an index):
Given n independent random varables Y(i) i=1,2,...,n each with E[Y(i)] = 0,
obviusly E[Y(1)+Y(2)+...+Y(n)] = |E[Y(1)+Y(2)+...+Y(n)]| = 0
Since the function "absolute value" is convex, for Jensen inequality we know that:
|E[Y(1)+Y(2)+...+Y(n)]| leq E[|Y(1)+Y(2)+...+Y(n)|]
But I'm pretty sure that this is the case for the two member being equal.
Do you agree?
All random variables have expectation = 0, therefore the expectation of their sum is = 0 and so is the expectation of the absolute value of their sum.
Am I correct?
implies the absolute integrability of , but is positive so the integral exists and is finite. But this last is .
Does not work is we allow improper integrals.