The random variable X is said to be stochastically larger than the random variable Y if P(Xz)or= P(Yz) for all real z with strict inequality holding for at least one z value. Show that this requires that the cdfs enjoy the following property: Fx(z)

Question

The random variable X is said to be stochastically larger than the random variable Y if P(X>z)>or= P(Y>z) for all real z with strict inequality holding for at least one z value. Show that this requires that the cdfs enjoy the following property: Fx(z)<or=Fy(z) for all real z with strict inequality holding for at least one z vale

zarkon · Answer

$$F_{X}(z)=P(X\le x),F_{Y}(z)=P(Y\le x)$$


if 

$$P(X>z)\ge P(Y>z)$$

then 

$$-P(X>z)\le -P(Y>z)$$
$$1-P(X>z)\le 1-P(Y>z)$$
$$P(X\le z)\le P(Y>\le z)$$
$$F_{X}(z)\le F_{Y}(z)$$