Independence and product probabilities

Two events are independent when their intersection probability is the product of their separate probabilities. Two random variables are independent when this factorization holds for every pair of measurable output sets. Equivalently their generated sigma-algebras are independent. For real variables, checking a suitable generating family such as pairs of lower half-lines is enough, with a measure-uniqueness argument extending the result. Checking just one pair of events is generally insufficient to establish independence of the full observations.

If an event B has positive probability, event independence is equivalent to the conditional probability of A given B being its unconditional probability. The product definition remains meaningful when B has probability zero, while the elementary ratio definition does not. This is one reason to use factorization as the foundation. Also independence is stable under measurable transformations performed separately on the variables: output questions about transformed values simply pull back to questions that were already independent.

The product of two probability measures assigns each measurable rectangle the product of its side masses and extends uniquely to the product sigma-algebra. On that product space the coordinate observations are independent by construction. For a nonnegative measurable function, Tonelli permits iterated integration without first proving a finite integral. For a signed function, Fubini's usual finite-valued conclusion needs absolute integrability. These theorems justify replacing a joint integral by successive averages; they do not license rearranging undefined differences of infinities.

For independent integrable real variables, the product is integrable and its expectation factors. Indeed the product integral of the absolute values is the product of their finite absolute expectations. For square-integrable variables this implies zero covariance. The converse fails: covariance tests only a particular pair of functions, while independence tests all measurable events. A symmetric variable and its square provide a familiar example: the covariance can vanish by symmetry even though the square is determined completely by the original variable.

A family is mutually independent when every finite subfamily and every choice of one event from each corresponding sigma-algebra satisfy the full product identity. Pairwise independence asks only about subfamilies of size two. These requirements agree for two variables but differ as soon as a third variable enters. To see the distinction, use two independent fair bits and let a third record their parity. Each pair looks independent, while all three together satisfy a deterministic relation.

The distinction has practical consequences. Pairwise independence suffices to eliminate covariance terms in the variance of a finite square-integrable sum. It does not justify multiplying an arbitrary collection of event probabilities or treating all coordinates as a product measure. Some limit theorems admit pairwise-independent variants, but such extensions need their own hypotheses and proofs. When a calculation invokes an independent sequence, interpret the default assumption as mutual independence of every finite subcollection, rather than merely an absence of visible pairwise associations.