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.

Thanks in advance.


Not necessarily.

Let X be 1,100 or 200 with probability 0.1,0.8,0.1 so E[X]=100.1
and Y be 1,100 or 200 with probability 0.2,0.6,0.2 so E[Y]=100.2
But, $E[log(X)]\approx 4.2>E[log(Y)]\approx 3.8$
Also, $E[\sqrt{X}]\approx 9.5>E[\sqrt{Y}]\approx 9.0$

  • $\begingroup$ But in this example X and Y are not defined in the same probabilistic space, because it has different probabilities. $\endgroup$ Feb 9 at 20:35
  • $\begingroup$ @user13761697 I think you have the wrong idea of what a probability space is $\endgroup$ Feb 9 at 21:36
  • $\begingroup$ The probabilities must be the same, you can change the value of $X(\omega)$ and $Y(\omega)$. These are my hypothesis. $\endgroup$ Feb 10 at 7:41

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