# First Order Stochastic Dominance relation of conditional distributions

Suppose there are two CDF's $$F$$ and $$G$$ over the same support, $$[0,1]$$, and assume that one first order stochastically dominates the other: $$\tag{FOSD 1}F\succsim_{FOSD}G$$ meaning that $$F(x)\leq G(x),~\forall x\in[0,1]$$.

Can we say this fact implies the dominance relation of the conditional distributions of the following? $$\tag{FOSD 2}F(x|x\geq y)\succsim_{FOSD} G(x|x\geq y),~\forall y\in[0,1]$$

Clearly, (FOSD $$2$$) implies (FOSD $$1$$) as we can just take $$y=0$$. However, can we guarantee that the converse also hold? or do we need something more to guarantee (FOSD $$2$$)?

• $F(x|x\ge y)$ does not make any sense. Maybe you mean $P(X\le x|X\ge y)$ where $X$ is a random variable having distribution $F$? Feb 15 '19 at 10:17
• That's correct. I meant the probability. Feb 15 '19 at 11:05