Realization (probability)

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In probability and statistics, realization, or observed value, of a random variable is the value that is actually observed (what actually happened). The random variable itself should be thought of as the process how the observation comes about. Statistical quantities computed from realizations are often called "empirical", as in empirical distribution function, empirical probability, or the empirical definition of sample.

Conventionally, random variables are denoted by uppercase letters while the realizations by the corresponding lowercase letters^[1]. Confusion and errors result when this important convention is not strictly observed.

In probability theory, random variable is a function X defined on a sample space Ω ^[2], and realization is its value. (In fact, a random variable cannot be an arbitrary function and it needs to satisfy some other conditions, but that is not important for the discussion here.) Elements of the sample space are thought of as all the different possibilities that might happen. Probability is a mapping that assigns numbers between zero and one to certain subsets of the sample space. Subsets of the sample space that contain only one element are called elementary events. The value of the random variable (that is, a function) X at a point ,

x = X(ω)

is called a realization of X.

[edit] References

1. ^ Samuel S. Wilks. Mathematical statistics. A Wiley Publication in Mathematical Statistics. John Wiley \& Sons Inc., New York, 1962.
2. ^ S. R. S. Varadhan. Probability theory, volume 7 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2001.