What probability density is used in Brownian motion?

[Earthland]

Homework Statement

I have a free Brownian particle and its coordinate is given as a function of time:

And its first moment, or mean, is given as

But what kind of probability density was used to calculate this first moment?

Homework Equations

I know that the first moment is calculated by using probability density

<x> = ∫x*W(x)dx from -infinity to +infinity.

X itself is dependent on t, so is it W(x) or W(x,t) or W(t)? I'm little lost on this one.

The Attempt at a Solution

I did find some kind of density function in regards to Brownian motion

But I can't see how this would yield the correct result. And still, must I integrate over x or t?

 

[Orodruin]

Just take the expectation value of your first expression (expectation values are linear) keeping in mind that the only actually random variable in it is the driving force $\zeta(t')$.