If X ≤ Y, then E(X) ≤ E(Y)

  • Thread starter kingwinner
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  • #1
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Main Question or Discussion Point

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!
 

Answers and Replies

  • #2
The theorem assumes that X and Y are defined on the same probability space [itex]\Omega[/itex]. [itex]X\leq Y[/itex] means [itex]X(\omega)\leq Y(\omega),\quad \forall\omega\in\Omega[/itex]. Actually, it would be enough to have [itex]X(\omega)\leq Y(\omega)[/itex] for P-almost all [itex]\omega\in\Omega[/itex], where P is the probability measure on [itex\Omega][/itex].
 
  • #3
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The theorem assumes that X and Y are defined on the same probability space [itex]\Omega[/itex]. [itex]X\leq Y[/itex] means [itex]X(\omega)\leq Y(\omega),\quad \forall\omega\in\Omega[/itex]. Actually, it would be enough to have [itex]X(\omega)\leq Y(\omega)[/itex] for P-almost all [itex]\omega\in\Omega[/itex], 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([itex]\omega[/itex]) < Y([itex]\omega[/itex]) for ALL [itex]\omega\in\Omega[/itex]?
 
Last edited:
  • #4
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).

[tex]
P(X\leq Y)=P\left(\omega\in\Omega: X(\omega)\leq Y(\omega)\right)
[/tex]

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 [itex]\Omega[/itex]. While on the left hand side...well...it is only a shorthand for the right side:smile:
 
  • #5
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Thanks! I love your explanations!
 
  • #6
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Two follow-up questions:

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

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 [itex]\Omega[/itex] (or almost all), right??
 
  • #7
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Two follow-up questions:

1) For P(X ≤ Y), do X and Y have to be defined on the SAME sample space [itex]\Omega[/itex]?
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 [itex]\Omega[/itex] (or almost all), right??
If they don't say a.e. or a.s. you can assume they mean for all [itex] \omega \in \Omega [/itex].
 

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