Independent Statistics

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

$$P(A\vert B) = P(A),$$ (1)

in which case the probability of $A$ and $B$ is just

$$P(AB)=P(A\cap B) = P(A)P(B).$$ (2)

Similarly, $n$ events $A_1$, $A_2$, ..., $A_n$ are independent Iff

$$P\left({\,\bigcap_{i=1}^n A_i}\right)= \prod_{i=1}^n P(A_i).$$ (3)

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

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