If X is a continuous Random Var with continuous density function $f$ and we have another Random Var $Y=F(X)$ where F is cumulative distribution function of X. What is expectation of Y?
I did this problem before coming across the accepted answer here and the way I did it seems a little bit different (or perhaps the answer's notation is throwing me off). Could someone double check the way I did it:
$E(Y) = E(F(X)) = \int_{-\infty}^{+\infty} F(x)f(x) \,dx$. Since f(x) is density function for X and F(X) is cumulative function, we have $f(x) = \frac{dF(X)}{dx} = F^{'}(x)$
so $\int_{-\infty}^{+\infty} F(x)f(x) \,dx = \int_{-\infty}^{+\infty} F(x)F^{'}(x) \,dx = .5*F(X)^2 |_{-\infty}^{+\infty} = .5*1 -0 = 1/2$. Is this a valid way to do this?
