Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}: \mathbb D^N \to\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ is either $\mathbb{R}$ or $\mathbb{Z}$ (i.e. reals or integers).
Can we ever have that
$\mathbb{E}_\psi \left(\underset{\mathbf{x}}{\arg\!\min}\; f_{\boldsymbol{\alpha}}(\mathbf{x})\right) = \underset{\mathbf{x}}{\arg\!\min} \; \mathbb E_\psi\left(f_{\boldsymbol{\alpha}}(\mathbf x)\right)$ for some choice of $\psi$ and/or $f$ ?
I am particularly interested in convex programming problems, that is, when $f$ is convex and only defined within a convex region, and specifically in LP problems, when $f$ is also a linear function, but an answer for any $f$ would suffice.
A stronger version of this question is when would we have LHS $=$ RHS in the Eq. above, but any examples of when LHS = RHS for a linear programming problem would already be very helpful.