Given the following data: $ x(t) = A + \omega(t) $ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$?
I know it could be proven to be: $ p(x;A) = C \exp\left(- \frac{\int (x(t) - A))^2 dt}{2\sigma^2}\right) $
Yet I don't know the reason and the formal proof.
Note:
Both RHS and LHS depends on the whole random process.