What does it mean for X and Y to be compared in terms of their expected values?

[kingwinner]

There is a theorem that says:
"Let X and Y be random variables. If X ≤ Y, then E(X) ≤ E(Y)."

But I don't really understand the meaning of "X ≤ Y". What does it mean?
For example, if X takes on the values 0,1,2,3, and Y takes on the values -1,2,5. Is X ≤ Y??

Any help is appreciated!

 

[Pere Callahan]

The theorem assumes that X and Y are defined on the same probability space $\Omega$. $X\leq Y$ means $X(\omega)\leq Y(\omega),\quad \forall\omega\in\Omega$. Actually, it would be enough to have $X(\omega)\leq Y(\omega)$ for P-almost all $\omega\in\Omega$, where P is the probability measure on [itex\Omega][/itex].

 

[kingwinner]

The theorem assumes that X and Y are defined on the same probability space $\Omega$. $X\leq Y$ means $X(\omega)\leq Y(\omega),\quad \forall\omega\in\Omega$. Actually, it would be enough to have $X(\omega)\leq Y(\omega)$ for P-almost all $\omega\in\Omega$, where P is the probability measure on [itex\Omega][/itex].

Thanks! Now I understand what X ≤ Y means in the theorem.

Consider a separate problem. How about X ≤ Y in the context of finding P(X ≤ Y)? In this case, do X any Y have to be defined as random variables with X($\omega$) < Y($\omega$) for ALL $\omega\in\Omega$?

 

[Pere Callahan]

No. In order to compute P(X ≤ Y), you have to take the probability of all omega such that X(omega) ≤ Y(omega). There might be other omega which do not satisfy this inequality but then they don't contribute to P(X ≤ Y).

$$P(X\leq Y)=P\left(\omega\in\Omega: X(\omega)\leq Y(\omega)\right)$$

You may have noticed that the probability measure P has strictly speaking two different meanings here. On the right hand side it is a function which takes as argument a subset of $\Omega$. While on the left hand side...well...it is only a shorthand for the right side

 

[kingwinner]

Thanks! I love your explanations!

 

[kingwinner]

Two follow-up questions:

1) For P(X ≤ Y), do X and Y have to be defined on the SAME sample space $\Omega$?

2) In order statistics, when they say X₍₁₎≤X₍₂₎≤...≤X_(n), they actually mean X₍₁₎(ω)≤X₍₂₎(ω)≤...≤X_(n)(ω) for each and for all ω E $\Omega$ (or almost all), right??

 

[Focus]

Two follow-up questions:

1) For P(X ≤ Y), do X and Y have to be defined on the SAME sample space $\Omega$?

They need to be on the same probability space. Having the same sample space is not enough.

2) In order statistics, when they say X₍₁₎≤X₍₂₎≤...≤X_(n), they actually mean X₍₁₎(ω)≤X₍₂₎(ω)≤...≤X_(n)(ω) for each and for all ω E $\Omega$ (or almost all), right??

If they don't say a.e. or a.s. you can assume they mean for all $\omega \in \Omega$.