Probability Theory 13 | Independence for Random Variables [dark version]

Independence for random variables is defined by checking whether the events created from their values behave independently—then that property can be expressed compactly using distribution functions. For two real-valued random variables X and Y on a probability space (Ω, 𝒜, P), independence means that for any real numbers x and y, the events {X ≤ x} and {Y ≤ y} are independent. Because these events come from pre-images of intervals under X and Y, the definition is grounded in the measurable structure of random variables: pre-images of sets like (−∞, x] generate σ-algebras, and independence requires that every pair of events drawn from the σ-algebras generated by X and by Y factorize in probability.

This event-based requirement translates into a clean statement about cumulative distribution functions. Independence holds exactly when the joint cumulative distribution function can be written as a product of the marginals: the probability of {X ≤ x, Y ≤ y} equals P(X ≤ x)·P(Y ≤ y). In notation, this is expressed as F_{X,Y}(x, y) = F_X(x)F_Y(y), where F_{X,Y} is the joint CDF (a “generalized” CDF because it depends on two inputs) and F_X, F_Y are the usual CDFs of X and Y. The practical takeaway is that independence can be verified by checking that the joint CDF factorizes into the product of the individual CDFs.

A standard situation where this factorization appears is when X and Y come from separate components of a product probability space. Suppose Ω = Ω₁ × Ω₂, with X depending only on the Ω₁ coordinate and Y depending only on the Ω₂ coordinate. Concretely, if Ω₁ represents the outcome of the first die and Ω₂ represents the outcome of the second die, then X can be defined as “the first die’s number” and Y as “the second die’s number.” Because X ignores the second coordinate and Y ignores the first, the events {X ≤ x} and {Y ≤ y} correspond to conditions on disjoint parts of the sample space, making the factorization—and thus independence—automatic.

The same logic extends beyond pairs to whole families of random variables. For an indexed collection {X_i}_{i∈I}, the family is independent if, for every finite subset J ⊂ I, the probability of simultaneously satisfying {X_j ≤ x_j for all j∈J} equals the product of the individual probabilities ∏_{j∈J} P(X_j ≤ x_j). This definition generalizes two-variable independence by requiring the same factorization property for every finite group of variables, with the understanding that the index set I may be infinite. The result is a scalable criterion: independence for many random variables is enforced by consistent product behavior across all finite subcollections.