A basic probability inequality

[forumfann]

I am confused on a basic probability inequality, could anyone help me on this:
If X>Y>t, then is P(X>t) larger or smaller than P(Y>t)?

Thanks

[HallsofIvy]

I am confused about this. You start by asserting that "X> Y> t". Given that, P(x> t) and P(Y> t) are both 1.0!

If you just mean that X and Y are random variables, requiring that X> Y, and t is some number, then, yex, P(X> t) is greater than or equal to P(Y> t) (not necessarily larger- they might be equal). If Y> t, then X> t follows from X> Y. But X> t may be true even if Y> t is not.

[statdad]

This may not be the point of the question about the r.vs, but: we say that the random variable Y is stochastically larger than the random variable X provided that


P(Y > t) \ge P(X > t), \quad \forall t


This idea is one way of discussing two-sample location problems.