I am reading the Zwanzig's book and I have a confusion about the average of the fluctuating force and its distribution.
As it says $F(t)$ is a random variable that means it has a probability distribution means the probability of getting the different amplitude of Fluctuating force at the same time $t=\epsilon$(say) is according to the distribution curve, that is the meaning of distribution curve.
Then it says the time average of the $F(t)$ over small time gap is zero,
From this relation, it is clear to me that the curve of $F(t)$ vs $t$ cross the $t=0$ axis many time over the small interval of time. But then He said that the distribution curve of $F(t)$ is Gaussian, How he came to this conclusion.
What is the relation between $F(t)$ vs t and the probability distribution curve of $F(t)$?