The zero-mean models have a constant mean and constant variance and shows no predictable trends or seasonality. Observations from a zero mean model are assumed to be **independent and identically distributed** (**iid**) and represent the random noise around a fixed mean, which has been deducted from the time series as a constant term.

Let us consider that *X _{1}, X_{2}, ... ,X_{n}* represent the random variables corresponding to n observations of a zero mean model. If

*x*are n observations from the zero mean time series, then the joint distribution of the observations is given as a product of probability mass function for every time index as follows:

_{1}, x_{2}, ... ,x_{n}

*P(X1 = x1,X2 = x2 , ... , Xn = xn) = f(X1 = x1) f(X2 = x2) ... f(Xn = xn)*Most ...