Statistical Independence

Related concepts

Principal Component Analysis (PCA)

Definition

Statistical independence is a concept in probability theory. Two events A and B are statistical independent if and only if their joint probability can be factorized into their marginal probabilities, i.e., P(A ∩ B) = P(A)P(B). If two events A and B are statistical independent, then the conditional probability equals the marginal probability: P(A|B) = P(A) and P(B|A) = P(B). The concept can be generalized to more than two events. The events A₁, …, A _n are independent if and only if $P(\bigcap_{i=1}^{n} A_i) = \prod_{i=1}^{n} P(A_i)$.

Theory

Two random variables X and Y are independent if and only if the events {X ≤ x} and {Y ≤ y} are independent for all x and y, that is, F(x, y) = F _X (x)F _Y (y), where F(x, y) is the joint cumulative distribution function and F _X and F _Y are the marginal cumulative distribution functions of X and Y, respectively. If X and Y are continuous random variables, then X and Y are independent if f(...