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Independent Statistics

Two variates $A$ and $B$ are statistically independent Iff the Conditional Probability $P(A\vert B)$ of $A$ given $B$ satisfies

P(A\vert B) = P(A),
\end{displaymath} (1)

in which case the probability of $A$ and $B$ is just
P(AB)=P(A\cap B) = P(A)P(B).
\end{displaymath} (2)

Similarly, $n$ events $A_1$, $A_2$, ..., $A_n$ are independent Iff
P\left({\,\bigcap_{i=1}^n A_i}\right)= \prod_{i=1}^n P(A_i).
\end{displaymath} (3)

Statistically independent variables are always Uncorrelated, but the converse is not necessarily true.

See also Bayes' Formula, Conditional Probability, Independence Complement Theorem, Uncorrelated

© 1996-9 Eric W. Weisstein