A random variable a function, which maps every outcome of a given random experiment to the set of real numbers, formally \[X:\cases{\Omega\mapsto\mathbb R\\A\mapsto X(A)}\]
If we have an experiment, in which the random variable \(X\) took the real value \(x\), then we call \(x\) the realization of \(X\).
The entirety of all possible realizations of \(X\) in the given experiment (formally the image of the function \(X(\Omega)\)) is called the population of \(X\). A population can be finite, or infinite, depending on whether the \(\Omega\) is finite or infinite. A sample is a finite number of realizations of \(X\). If the number of such realizations is \(n\), we say that we have a sample of size \(n\).
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