I'll try to explain more in detail: I have observations y(t), which are "known" (=assumed) to be of the previously presented integral form. Now, x(t) is the true value to be estimated, which is conventionally assumed to be some deterministic function. In such case it's easy to handle y. However, it is known that the observations are contaminated with some noise, so it would be natural to deal x(t) as a random variable (mu = true value) instead of a deterministic, and therefore also y(t) would be random as well(?). The next step will be to find ML-estimate for mu when only observations y(t) are known (x,mu,sigma unknown), but I'd like to get rid of the integral before searching (at least numerical) ML-estimate. If I can find a distribution for y, I could find ML-estimate for mu (and sigma), right?

The problem would of course be easily solved by adding the noise term OUTSIDE the integral, but I don't want to do that because I need to separate true variations of x and measurement equipment noise, while the observation probably contains both.

I don't know if this makes the problem even less understandable...