It is well known that expectation preserves convexity: If $f(x)$ is convex and $Y$ is a random variable, then $\mathbb E[f(x-Y)]$ is convex. This property arises in, for example, inventory theory.
I have not been able to find a good source to cite for this well-known fact. Can anyone suggest one?
(For what it's worth, Boyd and Vandenberghe's book proves another well known property, namely, minimization preserves convexity, but I don't think they prove it for expectation.)