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Let $X$ and $Y$ be any random variables defined in the same sample space. What do we mean exactly by $X>Y$?

Also, what do we mean $X>c$, where $c$ is a constant?

Thank you very much!

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    $\begingroup$ How are you defining a random variable? $\endgroup$ – anomaly Feb 11 '17 at 13:00
  • $\begingroup$ Thank you. Let $X$ and $Y$ be any random variables defined in the same sample space. $\endgroup$ – Probability is wonderful Feb 11 '17 at 13:16
  • $\begingroup$ I was asking whether you're defining a random variable as something vague like an "event" (which is standard for probability courses that don't talk about measure theory), or as a measurable function $f:\Omega \to \mathbb{R}$ (or $\mathbb{C}$, etc.) as below. $\endgroup$ – anomaly Feb 11 '17 at 13:40
  • $\begingroup$ I see. Sorry for my inaccuracy. I'd like to define a random variable as a measure function, like what you just said $\endgroup$ – Probability is wonderful Feb 11 '17 at 14:08
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Random variables are measurable functions from a measurable space $(\Omega, \mathcal A)$ to another measurable space $(G, \mathcal G)$. Usually the latter space is $(\mathbb R, \mathcal B(\mathbb R)$. So saying $X > Y$ means that, for every $\omega \in \Omega$, we have $$X(\omega) > Y(\omega)$$

Of course since $X(\omega), Y(\omega) \in G$, it is implied that the relation "$>$" is defined in $G$

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  • $\begingroup$ I see. Thank you very much! So, $X>Y$ implies that $P(X>Y)=1$ , right? $\endgroup$ – Probability is wonderful Feb 11 '17 at 14:10
  • $\begingroup$ @Probabilityiswonderful yes :) $\endgroup$ – Ant Feb 11 '17 at 14:12
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While @Ant's answer is correct there can be another meaning. "$X>Y$" can refer to some event, which can be formalized as $$ \{\omega:X(\omega)>Y(\omega)\}. $$ In which case $P(X>Y)$ can have any value.

You can understand $Y>c$ in the same way by considering $c$ a a (constant) random variable $(c(\omega)=c\ \forall \omega)$.

A simple example: throw two dice, observe the scores $X$ and $Y$. You can compute $P(X>Y)=5/12$.

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