Random variables X and Y are declared independent in the measure-theoretic sense. Which definition is fully general — applying to both discrete and continuous random variables without special-casing?
AThe correlation ρ(X, Y) = 0
BP(X = x, Y = y) = P(X = x) · P(Y = y) for all x, y — the joint probability factorization
CThe joint density factors as f_{X,Y}(x,y) = f_X(x) · f_Y(y) for all x, y
DThe sigma-algebras generated by X and Y are independent: P(A ∩ B) = P(A)P(B) for all A ∈ σ(X), B ∈ σ(Y)
The sigma-algebra definition (option D) is the fully general one. Option B (joint probability factorization) only applies to discrete variables. Option C (density factorization) only applies to continuous variables with a joint density. Option A (zero correlation) is necessary but not sufficient for independence — uncorrelated variables can be dependent. The sigma-algebra definition subsumes all cases: for discrete variables it reduces to option B, for continuous ones to option C, and handles general distributions without special-casing.
Question 2 Multiple Choice
Three sigma-algebras G₁, G₂, G₃ are pairwise independent: P(Aᵢ ∩ Aⱼ) = P(Aᵢ)P(Aⱼ) for all i ≠ j and appropriate events. Can we conclude they are mutually independent?
AYes — pairwise independence of all pairs implies mutual independence
BYes — for exactly three sigma-algebras, pairwise independence is equivalent to mutual independence
CNo — mutual independence also requires P(A₁ ∩ A₂ ∩ A₃) = P(A₁)P(A₂)P(A₃) for all event choices, which pairwise independence does not guarantee
DNo — mutual independence requires the sigma-algebras to be generated by continuous random variables
Pairwise independence does not imply mutual independence. A classic counterexample: flip two fair coins; let X₁, X₂ be the outcomes and X₃ = X₁ XOR X₂. Then σ(X₁), σ(X₂), σ(X₃) are pairwise independent, but P(X₁=H, X₂=H, X₃=T) = 1/4 ≠ P(X₁=H)P(X₂=H)P(X₃=T) = 1/8. Mutual independence requires the product rule to hold for every finite subset — not just pairs — and this is a strictly stronger condition.
Question 3 True / False
Two random variables X and Y are independent (in the measure-theoretic sense) if and only if the sigma-algebras they generate, σ(X) and σ(Y), are independent.
TTrue
FFalse
Answer: True
This is the definition: measure-theoretic independence of random variables IS defined as independence of their generated sigma-algebras. σ(X) contains all events of the form {X ∈ B} for Borel sets B — everything X could possibly reveal about the underlying outcome. Independence of σ(X) and σ(Y) means that no event in one sigma-algebra provides any probabilistic information about any event in the other, which is exactly what independence of random variables should mean.
Question 4 True / False
If three events A, B, C satisfy pairwise independence (P(A∩B) = P(A)P(B), P(A∩C) = P(A)P(C), P(B∩C) = P(B)P(C)), then they are mutually independent.
TTrue
FFalse
Answer: False
Pairwise independence of events does not imply mutual independence. Mutual independence additionally requires P(A ∩ B ∩ C) = P(A)P(B)P(C). Standard counterexamples (e.g., the Bernstein coins construction) show that the triple product condition can fail even when all pairwise conditions hold. The distinction matters in probability theory: proofs requiring mutual independence (such as some CLT extensions) cannot rely on pairwise independence alone.
Question 5 Short Answer
Why does the measure-theoretic definition of independence via sigma-algebras provide a unifying framework that the elementary event-based definition (P(A ∩ B) = P(A)P(B)) does not? What structural gap does it close?
Think about your answer, then reveal below.
Model answer: The elementary definition handles pairs of specific events but cannot be directly extended to random variables (which generate infinitely many events) or to sequences of random variables without special-casing discrete vs. continuous distributions. The sigma-algebra σ(X) captures ALL information X could reveal about ω, so requiring σ(X) and σ(Y) to be independent simultaneously enforces the product rule for every possible event pair — covering discrete factorization, density factorization, and arbitrary distributions in one statement. This unified definition is what allows the laws of large numbers and the CLT to be stated and proved for i.i.d. sequences without case analysis.
The sigma-algebra framework also handles the extension to countably many variables naturally: a sequence X₁, X₂, ... is i.i.d. if σ(X₁), σ(X₂), ... are mutually independent (product rule for all finite subcollections) and all Xᵢ have the same distribution. Without sigma-algebras, making this precise for uncountable probability spaces is awkward.