Expected value and nonnegative random variable

[webbster]

Hi All,

i got a short question concerning the ev of a monotone decreasing function.


when i got a nonnegative random variable t, then its ev (with a continuous density h(.)) is given by
E(t)=[int](1-F(t))dt
Then if v is a nonpositive random variable, is its ev given by
E(v)=-[int](1-F(v))dv
?
Hence,
i got that the ev of a monotone increasing function g(x) is:
E(g(x))=[int]g'(x)(1-F(x))dx

Now, let b(x) denote a monotone decreasing function. Therefore: z(x)=-b(x) is a monotone increasing function.
Am I correct, that it got the ev of b(x) by
E(b(x))=-E(z(x))
and thus
E(b(x))= - [int]z'(x)(1-F(x))dx
?

any thoughts are highly appreciated!

thanks a lot!

 

[EnumaElish]

What is F(.)? How is it related to h(.)? Why not use E[v] = [int] v h(v) dv?
http://en.wikipedia.org/wiki/Expected_value#Mathematical_definition

 

[webbster]

What is F(.)? How is it related to h(.)? Why not use E[v] = [int] v h(v) dv?
http://en.wikipedia.org/wiki/Expected_value#Mathematical_definition

oh i am sorry... just a typo.

F(.) should be H(.) and represent the cdf to the pdf h(.)...