I am trying to understand if the following statement is true, or the conditions under it is satisfied. Let $M,N$ and $X>0$ be random variables. If the following inequality holds for any concave non-decreasing function $u$ \begin{equation} \mathbb{E}[u(N)]\leq\mathbb{E}[u(M)] \end{equation} Then \begin{equation} \mathbb{E}[u(NX)]\leq\mathbb{E}[u(MX)] \end{equation} Note that there is no an assumed relation between $N$ and $M$.
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$\begingroup$ what about the relationship between $X$ and $N,M$? $\endgroup$– AlbertoCommented Jun 17 at 7:11
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Define $N$ and $M$ as in a similar question asked by you before, so that $E[u(N)] \leq E[u(M)]$ holds for any concave non-decreasing function.
Let $X = N + 2 > 0$, then \begin{align*} & E[u(NX)] = E[u(N^2 + 2N)] = E[u(2N + 1)] = \frac{1}{2}u(-1) + \frac{1}{2}u(3), \\ & E[u(MX)] = u(0). \end{align*}
Clearly, there are countless concave non-decreasing functions $u$ such that $ u(-1) + u(3) > 2u(0)$ -- perhaps the most trivial one is $u(x) = x$.
