I am confused about what is means to take the expected value of a CDF.
Let $X$ be a random variable with cdf $F_x(x)$, and assume that $X$ is a continuous random variable. What is $E[F_x(X)]$ and $var(F_x(X))$.
I assume this is similar to just taking the expectation of a function of a random variable, so I get something like this to start out.
$E[F_x(X)] = \int F_x(x)f_x(x)dx$
where $f_x(x)$ is the pdf of $X$, but I am not sure what this even means or where to go next. I understand expected value of a random variable, but what does it mean to take the expected value of a CDF. I am sorry if this has been answered, but I am not able to find help on this anywhere.
We make the substitution $F_x(u)=t$. Then $f_x(u)du=dt$. Note $t$ ranges from $0$ to $1$ $$E[F_x(X)] = \int_{-\infty}^{\infty} F_x(u)f_x(u)du= \int_0^1 t dt =\frac12$$ Similarly $$E[F_x(X)^2] = \int_{-\infty}^{\infty} F_x(u)^2f_x(u)du= \int_0^1 t^2 dt =\frac13$$ Hence $Var[F_x(X)]=\frac1{12}$.