= Solution
The <wave function> is a <probability> amplitude: by the <Born rule>, $|\psi(x)|^2dx$ is the <probability> of a position measurement in an interval of width $dx$. For a normalized bound state $\int|\psi|^2dx=1$, the <expected values> are
$$
\boxed{\langle x\rangle=\int x|\psi(x)|^2\,dx,\qquad
\langle x^2\rangle=\int x^2|\psi(x)|^2\,dx.}
$$
The integrals run over the allowed domain. If the given state is not normalized, divide both by $\int|\psi|^2dx$. Its <variance> is $\langle x^2\rangle-\langle x\rangle^2$; neither $\psi$ itself nor its overall phase is a directly measured position <probability>.
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