Questions tagged [independence]

For events: Two events $A$ and $B$ are independent if $$P(A\cap B)=P(A)P(B)$$ More generally, a family $\mathscr F$ of events is independent if, for every finite number of distinct events $A_1$, $A_2$, $\ldots$, $A_n$ in $\mathscr F$, $$P\left(\bigcap_{i=1}^nA_i\right) =\prod_{i=1}^nP(A_i)$$

Two collections of events (for example, two $\sigma$-algebras) $\mathscr F$ and $\mathscr G$ are mutually independent (or simply, independent) if every $A$ in $\mathscr F$ and every $B$ in $\mathscr G$ are independent.

More generally, some collections $\mathscr F_i$ of events, indexed by some finite or infinite set $I$, are mutually independent (or simply, independent) if, for every finite subset $\\{i_1,i_2,\ldots,i_n\\}$ of $I$ and every event $A_k$ in $\mathscr F_{i_k}$, the family $\\{A_1,\ldots,A_n\\}$ is independent.

For random variables: Two random variables $X$ and $Y$ (defined on the same probability space) are independent if their $\sigma$-algebras $\sigma(X)$ and $\sigma(Y)$ are (mutually) independent.

In particular, 2 events $A$ and $B$ are independent if and only if the indicator random variables $1_A$ and $1_B$ are independent.

More generally, a family $\mathscr X$ of random variables (defined on the same probability space) is independent if, for every finite sub-family $\\{X_1,X_2,\ldots,X_n\\}$ of $\mathscr X$, the $\sigma$-algebras $\sigma(X_{1})$, $\sigma(X_{2})$, $\dots$, $\sigma(X_{n})$ are (mutually) independent.

An idea akin to probalistic independence exists in mathematical logic:

A sentence $\sigma$ is independent of a theory T if T neither proves nor refutes $\sigma$. Synonymously, $\sigma$ is said to be undecidable from T.

A theory T is independent if no axiom in T is provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable.