A random variable is a mapping from a sample space to real numbers $\Omega \rightarrow \mathrm{R}$ At a certain point in most probability courses, we don't see the sample space, but it's always there, lurking in the background. For example: Let $\Omega = \{(x,y); x^2 + y^2 \leq 1\}$ be the unit disc. Consider drawing a point "at random" from $\Omega$. Outcome: $\omega = (x,y)$. Examples of random variables: $X(\omega) = x$, $X(\omega) = y$, $Z(\omega) = x + y$
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