I would like to calculate conditional expectation $E[X|A]$, where $A$ is a set, only from the characteristic function $\phi(\omega)$ of a random variable $X$. How can I do this?
Since the characteristic function describes the density function completely, I should be able to do everything at the frequency domain but I dont know how it can be done. If there is no conditioning then, the result is simply the derivative of the characteristic function.
I also wonder how to calculate $$\int_{-\infty}^A f(t)\mathrm{d}t$$ from the chracteristic function $\phi(\omega)$ without going back to the density domain.
Thanks alot...
NOTES:
I found a solution to the second part of my question from
$$F_X(x)=\frac{1}{2}+\frac{1}{2\pi}\int_0^\infty \frac{e^{iwx}\phi_X(-w)-e^{-iwx}\phi_X(w)}{iw} \mathrm{d}w$$ with $F_X(A)$