# Expected value of two random variables and an increasing function.

Let $$X,Y$$ two real random variables defined over the same probabilistic space with $$X(\omega),Y(\omega)\geq 0$$ for every $$\omega\in\Omega$$ and such that $$\text{E}[X]\leq\text{E}[Y]$$. Let $$g : [0,\infty)\rightarrow [0,\infty)$$ be an increasing function. I wonder to know if $$\text{E}[g(X)]\leq\text{E}[g(Y)]$$ or find a counterexample. I was thinking in the function $$g(x) = \sqrt{x}$$, but i'm pretty lost.

But, $$E[log(X)]\approx 4.2>E[log(Y)]\approx 3.8$$
Also, $$E[\sqrt{X}]\approx 9.5>E[\sqrt{Y}]\approx 9.0$$
• The probabilities must be the same, you can change the value of $X(\omega)$ and $Y(\omega)$. These are my hypothesis. Feb 10 at 7:41