A random variable X is distributed in [0, 1]. Mr. Fox believes that X follows a distribution with cumulative density function (cdf) $F : [0, 1]\rightarrow [0, 1]$ and Mr. Goat believes that X follows a distribution with cdf $G : [0, 1] \rightarrow [0, 1]$. Assume F and G are differentiable, $F\neq G$ and $F(x) \leq G(x)$ for all $x\in [0, 1]$ . Let $E_{F} [X]$ and $E_{G}[X]$ be the expected values of X for Mr. Fox and Mr. Goat respectively. Which of the following is true?
(a)$E_{F}[X] \leq E_{G}[X]$
(b) $E_{F}[X] \geq E_{G}[X]$
(c) $E_{F}[X] = E_{G}[X]$
(d) None of the above.
The solution to this is: http://discussion-forum.2150183.n2.nabble.com/file/n7588025/sol.png
Which i find extremely difficult. Can someone provide me with an easier and more understandable possible solution? Thankyou!
The solution looks difficult because you need to first derive a key identity: for any nonnegative random variable $X$, the following holds:
$$E[X]=\int_0^\infty(1-F(x))dx.$$
That's the first part of the derivation. Once you have this the rest will follow.