Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point $\xi_0 \in \mathbb{R}$ and define $$f(x)=\chi_{\{\lvert x-x_0\rvert <\lambda_x\}}\mathscr{F}^{-1}\left(\chi_{\{\lvert \xi-\xi_0\rvert<\lambda_{\xi}\}}\right).$$ Question. What can we say about the expectation values of position $$\langle f|x|f\rangle=\int_{-\infty}^{\infty}x \lvert f(x)\rvert^2\, dx$$ and of momentum $$\langle f|p|f\rangle=\int_{-\infty}^{\infty}\xi \left\lvert \hat{f}(x)\right\rvert^2\, d\xi?$$ (Optional) What about the variances?
My guess is that the average position is $x_0$ and that the average momentum is $\xi_0$ regardless of the lenghtscales. Those will enter into the variance but I cannot predict the exact functional dependence.