Definition
A random variable X in the school formulation is a numerical function on the sample space Ω of one trial: to each outcome ω a number X(ω) is assigned.
A discrete random variable takes a countable set of isolated values (often a finite set x1,…,xk). A continuous one (in idealized models) may take values on a whole interval — for example, time until the first event or the coordinate of a point chosen at random.
“Random” in the name — because of the random outcome ω of the experiment, while X simply encodes it as a number
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