# Definition: Random Variable, Realization, Population and Sample

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 $$n$$.

Explanations: 1

Definitions: 1 2 3 4
Explanations: 5
Proofs: 6 7 8 9 10
Propositions: 11 12 13 14

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### References

#### Bibliography

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