Independence

Discrete Case

Two random variables $X$ and $Y$ are said to be independently distributed if

$$Pr(Y=y|X=x)=Pr(Y=y)$$

Since the conditional distribution implies that

$$Pr(Y=y|X=x)=\frac{Pr(X=x,Y=y)}{Pr(X=x)}$$

That is

$$Pr(X=x,Y=y)=Pr(X=x)Pr(Y=y)$$

Then the $X$ and $Y$ are independent.

Notice that all the conditions must satisfy, or the Independence do not exist.

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Continuous Case

$Y$ is independent of $X$ if

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