# About

Tag Info

For questions involving the notion of independence of events, of independence of collections of events, or of independence of random variables. Use this tag along with (probability) or (probability-theory). Do not use for linear independence of vectors and such.

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.

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.

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.