Independent Variables

If the joint distribution of two variables factorizes into the product of the marginals, such that

$$p(X,Y)=p(X)p(Y)$$

then $X,Y$ are said to be independent. From the product rule, this extends to give us:

$$p(Y|X)=p(Y)$$

Example of independent events: successive flip of a coin; previous flips has no effect on the future ones.