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| Title of object: |
independent |
| Canonical Name: |
Independent |
| Type: |
Definition |
| Created on: |
2001-12-03 23:57:37 |
| Modified on: |
2006-10-04 18:16:21 |
| Classification: |
msc:60A05 |
Revision comment (for changes between this and next version):
| Changes for correction #10956 ('mutual independence'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
In a probability space, we say that the random events $A_1,\dots,A_n$ are
\emph{independent} if
$$ P(A_1\cap A_2\dots\cap A_n) = P(A_1)\dots P(A_n). $$
An arbitrary family of random events is independent if every finite subfamily is independent.
The random variables $X_1,\dots,X_n$ are independent if, given any Borel sets $B_1,\dots,B_n$, the random events $[X_1\in B_1],\dots,[X_n\in B_n]$ are independent. This is equivalent to saying that
\[F_{X_1,\dots,X_n} = F_{X_1}\dots F_{X_n}\]
where $F_{X_1},\dots, F_{X_n}$ are the distribution functions of $X_1,\dots, X_n$, respectively, and $F_{X_1,\dots,X_n}$ is the joint distribution function. When the density functions $f_{X_1},\dots,f_{X_n}$ and $f_{X_1,\dots,X_n}$ exist, an equivalent condition for independence is that
\[f_{X_1,\dots,X_n} = f_{X_1}\dots f_{X_n}.\]
An arbitrary family of random variables is independent if every finite subfamily is independent. |
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