In order to change the order of integration, you must remove the from the bound. Here is how:
....
.... .
(I denote by an indicator function: it is equal to 1 if and to 0 otherwise)
I've got a problem that I don't know how to solve. The question: Suppose that Y is a continuous random variable with density f(y) that is positive only if y> or equal to zero. If F(y) is the distribution function, show that
The book also gives a hint: If y > 0, and
Exchange the order of integration to obtain the desired result.
I've tried to change the integration order like this:
but how do I go on from there?
Thanks in advance.
Consider (for some fixed ) . The function that is being integrated is equal to when (because the indicator equals 1), and it is equal to when (because the indicator equals 0). So it simplifies into: (we integrate only on the values of where the integrand is not zero, that is for greater than ). Is it clear? I use this operation twice: once to remove the from the integration sign, and once to put back a .
There is also a "visual" way to understand this exchange in the order of integration. Draw the set of all such that (with the -axis as the horizontal axis): this is the part of the plane delimited by the positive -axis and the half-line , . When you compute , you integrate the function along the vertical slices corresponding to fixed 's, and then you "sum" the slices together. These slices are bounded: from 0 to . And, by Fubini Theorem, the value of the integral does not change if you sum the horizontal slices instead: for each , the slice goes from to . So that the integral becomes: . That's clearer when drawn on a paper actually, but I hope you got it anyway.