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Definition 13   Two random variables $ X$ and $ Y$ are independent if:

$\displaystyle P(a\le X\le b,c\le X\le d) = P(a\le X \le b) P(c\le Y \le d) $

for all $ a,b,c,d$

Remark: If $ X$ and $ Y$ independent then:

$\displaystyle P(a\le X\le b\vert c\le Y\le d) = P(a\le X \le b) $

In other words, information about $ Y$ does not effect the probability of $ X$.

Eran Borenstein