Independent random variables

In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

The concept of independence extends to dealing with collections of more than two events or random variables, in which case the events are pairwise independent if each pair are independent of each other, and the events are mutually independent if each event is independent of each other combination of events.

Two events A and B are independent (often written as ${\displaystyle A\perp B}$ or ${\displaystyle A\perp \!\!\!\perp B}$) if their joint probability equals the product of their probabilities:

Why this defines independence is made clear by rewriting with conditional probabilities:

and similarly

Thus, the occurrence of B does not affect the probability of A, and vice versa. Although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if P(A) or P(B) are 0. Furthermore, the preferred definition makes clear by symmetry that when A is independent of B, B is also independent of A.

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