## Independence

Finally, we shall define the notion of independence between two continuous random variables.

DEFINITION 3.4.6 Continuous random variables X and Y are said to be independent if f{x, y) = fix)fiy) for all x and y.

This definition can be shown to be equivalent to stating

Fix і ^ X ^ X2, yi ^ Y — у2) = P(xi < X ^ X2)Piyi — Y ^ yfj

for all X], x%, yi, jo such that x < x%, yi ^ y^- Thus stated, its connection to Definition 3.2.3, which defined independence for a pair of discrete random variables, is more apparent.

Definition 3.4.6 implies that in order to check the independence between a pair of continuous random variables, we should obtain the marginal densities and check whether their product equals the joint density. This may be a time-consuming process...

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