For nonnegative random variables, expectation is defined to be the supremum of all expectations of simple random variables $A$ that satisfy $A\leq X$. (For simple random variables, $E(A)=\sum_jc_jP(C_j)$ where $c_j$'s are nonnegative constants and $C_j$'s are disjoint events in $\Omega$.)
Let $X$ be a nonnegative real-valued random variable, how can we show that $\mathrm{Var}(\min(X,y))$ is increasing in $y$? ($y$ is a constant.)
I thought that the proof is probably based on definition of expectation and I would need compare $\mathrm{Var}(\min(X,y_1))$ and $\mathrm{Var}(\min(X,y_2))$ for $y_1\leq y_2$ and probably start from some simple RVs.
Thank you for any help!