Explanation: Why is a random variable neither random, nor variable?

(related to Definition: Random Variable, Realization, Population and Sample)

The term "random variable" is very common in stochastic and statistics. However, it is somehow misleading, since a random variable \(X\) is neither random, nor it is variable. Why? Because it is a function. A function has always to be very deterministic, and well-defined. For instance, if we get the same outcome \(A\) of a random experiment twice, we can be sure that the real value of our random variable \(X(A)\) will be twice the same!

Thus, is is important to understand that random variables are deterministic, but their realizations (i.e. their possible values depending on the random outcomes of an experiment) are random.

Another important thing to understand is that random variables are different from probabilities. A random variable \(X\) always describes some numerical property that outcomes of our random experiments may have, rather than the probability it happens. However, we can talk about the probability of \(X\) having a special value, e.g.

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  1. Hedderich, J.;Sachs, L.: "Angewandte Statistik", Springer Gabler, 2012, Vol .14