Help with integral (inner product of stochastic and deterministic process)

Question

i need to calculate an integral of the form
$$ X = \int_0^T w(t) \sin (\omega t) dt $$ where $w(t)$ is a stochastic normal process (white noise), $\sin(\omega t)$ is deterministic. How do I do that?

The result is not important -it's just an example-, but the method, the how, is. The result is obviously random; what I'm trying to find is the pdf of X.

Edit: in this particular case, suppose $T = \frac{2\pi n}{w}$, so the integral is over a natural number of periods. How can I prove normality of $X$? If $X$ is normal, I would try something like the Parseval identity to calculate $\sigma$, but it has been a long time since I did this kind of operations. Ideas?

Answer 1

You may try

$$I(\omega)=\left(\int_0^T w(t) \sin (\omega t) dt\right)^2=\int_0^T \int_0^T w(t)w(s) \sin (\omega t) \sin(\omega s) dtds$$ Let $\left< X \right>$ denote the stochastic average.

Then $$\left<I(\omega)\right>=\int_0^T \int_0^T \left<w(t)w(s)\right> \sin (\omega t) \sin(\omega s) dtds$$

Now we assume that:

$$\left<w(t)w(s)\right>=f(|s-t|)$$

then

$$\left<I(\omega)\right>=\int_0^T \int_0^T f(|t-s|) \sin (\omega t) \sin(\omega s) dtds$$