The mean function of a time series process is defined as $$\mu_{xt}=E[X_t] $$ provided that it exists.
Example 1: If \( w_t \) denotes a white noise series, then If \( E(w_t)=0 \) for all \( t \).
Example
2:Consider the random walk with drift model $$x_t= \delta _t+
\frac{1}{n} \sum\limits_{j = 1}^t {{w_j}}, t=1,2, ...$$
Because\(
E(w_t)=0 \) for all \( t \) and \(\delta _t\) is a constant we have
$$ \mu_{xt}=E[X_t]=\delta _t+\sum\limits_{j = 1}^t {{w_j}}=\delta _t\
$$which is a straight line with slope \( \delta \).