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A variable whose value is determined by a process that we cannot predict. Random variables are usually denoted by capital letters. Eg if we roll a die, and let X=the number that comes up on the die, then X is a random variable.
We can often say that a particular random variable has a particular distribution, which is some set of information about that probabilities that the variable will take certain values.
A discrete random variable can only take values from a given discrete set (often the whole numbers); a continuous random variable can be any real number in a given range.

A function X : Ω→ R where Ω is the sample space. If Ω is finite or countable then X is said to be a discrete random variable.