First-Order Stochastic Dominance

Consider two cumulative distribution functions $F(x)$ and $G(x)$ for $x\in[a,b]$ where $G(x)$ has the first-order stochastic dominance over $F(x)$. That is, $F(x)>G(x)$ for all $x\in(a,b)$. We assume $a<0$ and $b>0$. Let $f(x)$ and $g(x)$ be the probability density function of $F(x)$ and $G(x)$ respectively.

Suppose the expected value of $x$ under $F(x)$ is positive: $$\int_{a}^{b}xf(x)dx=\int_{a}^{0}xf(x)dx+\int_{0}^{b}xf(x)dx>0.$$

Under this condition, does $f(x)-g(x)>0$ always hold in any interval of $0<x<b$?

Graphical Expression of the Question is Here.

• Thank you for your comments. Could you tell me how I can edit my question? Should I delete this question and post a new question? Jul 25 '18 at 16:01
• Please don't delete the question. There is a link that allows you to edit the question just below it. (It is just above and to the left of the box that shows your name.) Jul 25 '18 at 16:02
• Also, your definition of FOSD seems much stronger than the usual definition. Is that intentional? Jul 25 '18 at 16:04
• Yes, it is intentional. Thank you for pointing it out. Jul 25 '18 at 16:05

Suppose $a=-1$ and $b=2$, that $F$ is the uniform distribution in the interval $[-1,2]$, and that $G$ is the uniform distribution in the interval $[0,2]$. Clearly the expected value of $x$ under $F$ is positive but $$g(x)=\frac{1}{2}>\frac{1}{3}=f(x)\quad\text{for all}\;x\in[0,b].$$