Let $Y$ be a random variable. Define $f:\mathbb{R} \rightarrow \mathbb{R}$ with $f(x) = var(max(x, Y))$. Is $f$ convex?
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If $Y$ isn't degenerate, in general it is not true. – user251257 Oct 18 '16 at 22:37
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Consider $Y$ uniformly distributed on ${0,1}$. – user251257 Oct 18 '16 at 22:57
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@user251257 Thanks for your comment. You're right. Can you put that as an answer? – Mehdi Jafarnia Jahromi Oct 19 '16 at 01:01