Definition
Random Variable
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~Probability Space).
A random variable is a measurable function $X : \Omega \to \mathbb{R}$, meaning $X^{-1}(B) \in \mathcal{F}$ for every Borel set $B \subseteq \mathbb{R}$.
The law (or distribution) of $X$ is the pushforward measure $\mathbb{P}_X(B) := \mathbb{P}(X^{-1}(B))$ on $\mathbb{R}$.
Depends on
Dependency Graph
flowchart LR
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nbbb6ebd1["Probability Space"]
n3eb4aae1["Random Variable"]:::current
n09316b14["Expectation"]
n699b1d18["Independence"]
nccabff62["Stochastic Process"]
nbbb6ebd1 --> n3eb4aae1
n3eb4aae1 --> n09316b14
n3eb4aae1 --> n699b1d18
n3eb4aae1 --> nccabff62
click nbbb6ebd1 "../objects/bbb6ebd1.html" "_self"
click n09316b14 "../objects/09316b14.html" "_self"
click n699b1d18 "../objects/699b1d18.html" "_self"
click nccabff62 "../objects/ccabff62.html" "_self"