I'm studying about Kalman filter from the book of "optimal state estimation" and I have one exercise where I need to prove the following equation:
$$\frac{d}{dt}(E[x])=E\left[\frac{dx}{dt}\right].$$
I'm not quite sure how to approach this problem, because it is not clear to me are we taking the expectations with respect to what random variables? Is it the random variable $x$, or is the $t$ here also a random variable? First, I assumed the expectations are taken w.r.t to $x$, so I started to expand the left side of this equation first:
$$\frac{d}{dt}(E[x]) = \frac{d}{dt}\left(\int x\,f(x)\,dx\right)=\int \frac{d}{dt}\left(x\,f(x)\right)\,dx=\int \left[\frac{dx}{dt}f(x)+x\frac{df}{dx}\frac{dx}{dt}\right]\,dx$$
and then I looked at the right side of the equation:
$$E\left[\frac{dx}{dt}\right] = \int \frac{dx}{dt}\,f(x)\,dx$$
so now I'm stuck a little bit. If I take the expectations w.r.t $t$ I get:
$$\frac{d}{dt}(E[x]) = \frac{d}{dt}\left(\int x\,f(t)\,dt\right)=\int \frac{d}{dt}\left(x\,f(t)\right)\,dt=\int \left[\frac{dx}{dt}f(t)+x\frac{df}{dt}\right]\,dt$$
$$E\left[\frac{dx}{dt}\right] = \int \frac{dx}{dt}\,f(t)\,dt.$$
or should I take the expectations w.r.t to different random variables? Another point of confusion: are the differential coefficients constant in this problem? Or should I treat them as functions of $t$ (that is if expectations are taken w.r.t. $x$)?
