I am studying time series and came across the Mean function which the textbook defines as: $$ \mu_{xt}=E(X_t)=\int_{-\infty}^{\infty}xf_t(x)dx $$ I don't understand what this function does. I looked at the Mean of a function page but the formula looks different $$ \bar{f}={\frac {1}{b-a}}\int _{a}^{b}f(x)dx $$ Are these two functions the same?
What is the intuition of the Mean function and why is is important to know the mean of a time series function?
This is not the mean of the function, this is the expected value (https://en.wikipedia.org/wiki/Expected_value) of $X_t$. Think of it as the weighted average of $X_t$, where $f_t$ are the weights.