Realization (probability)

In probability and statistics, a 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 without deploying a statistical model are often called "empirical", as in empirical distribution function or empirical probability.

Conventionally, upper case letters denote random variables; the corresponding lower case letters denote their realizations.^[1] Confusion results when this important convention is not strictly observed.

In more formal probability theory, a random variable is a function X defined from a sample space Ω to a measurable space called the state space.^[2] If an element in Ω is sent to an element in state space by X, then that element in state space is a realization. (In fact, a random variable cannot be an arbitrary function and it needs to satisfy another condition: it needs to be measurable.) Elements of the sample space can be thought of as all the different possibilities that could happen; while a realization (an element of the state space) can be thought of as the value X attains when one of the possibilities did 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, the function) X at a point ω ∈ Ω,

x = X(ω)

is called a realization of X.

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.