In a similar way we can also define expectation values of functions $f(x,t)$ as $$\begin{equation*} \langle f(x,t)\rangle = \int_{-\infty}^{\infty}f(x,t)w(x,t)dx. \end{equation*}$$ The normalization condition $$\begin{equation*} \int_{-\infty}^{\infty}w(x,t)dx=1 \end{equation*}$$ imposes significant constraints on $w(x,t)$.