vault backup: 2026-04-21 17:54:23

realization.md

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+---
+title: Realization
+tags: []
+---
+# Realization
+
+> In [[probability]] and [[statistics]],
+> a **realization** or **observation**
+> (also called **observed value**)
+> of a random variable or random element
+> is the value that is actually observed or measured.[^1]
+
+[^1]: [Realization (probability) - Wikipedia](https://en.wikipedia.org/wiki/Realization_\(probability\))
+
+> In probability theory,
+> a random variable is a function $X$
+> defined from a sample space $\Omega$ to the state space.
+
+> If an element in $\Omega$ is mapped to an element in state space by $X$,
+> then that element in state space is a realization.
+> That is, at a point $\omega \in \Omega$,
+> the value $x = X(\omega)$
+> is called a realization of $X$