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

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Discussion Overview

The discussion revolves around the comparison of random variables X and Y in terms of their expected values, specifically focusing on the meaning of the notation "X ≤ Y" and its implications in probability theory. Participants explore the conditions under which this comparison holds and its relevance in calculating probabilities related to these variables.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of "X ≤ Y" in the context of the theorem stating that if X ≤ Y, then E(X) ≤ E(Y), providing an example with specific values for X and Y.
  • Another participant clarifies that "X ≤ Y" means X(ω) ≤ Y(ω) for all ω in the same probability space Ω, noting that it could also hold for almost all ω with respect to the probability measure P.
  • A follow-up question is posed regarding the computation of P(X ≤ Y), asking if X and Y must be defined such that X(ω) < Y(ω) for all ω in Ω.
  • A response indicates that to compute P(X ≤ Y), one must consider all ω such that X(ω) ≤ Y(ω), and that other ω not satisfying this inequality do not contribute to the probability.
  • Further questions are raised about whether X and Y need to be defined on the same sample space for P(X ≤ Y) and the interpretation of order statistics in terms of ω in Ω.
  • It is asserted that X and Y must be on the same probability space, and that in order statistics, the inequalities are meant to hold for all ω in Ω unless stated otherwise.

Areas of Agreement / Disagreement

Participants generally agree on the need for X and Y to be defined on the same probability space for the comparisons and calculations discussed. However, there are nuances regarding the interpretation of "almost all" and the specifics of order statistics that remain open to further clarification.

Contextual Notes

Some participants note the distinction between having the same sample space and being on the same probability space, indicating a potential area of confusion. The discussion also touches on the implications of "almost all" in the context of probability measures.

kingwinner
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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!
 
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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].
 
Pere Callahan said:
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?
 
Last edited:
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).

<br /> P(X\leq Y)=P\left(\omega\in\Omega: X(\omega)\leq Y(\omega)\right)<br />

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:smile:
 
Thanks! I love your explanations!
 
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??
 
kingwinner said:
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.
 

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