Pairwise independence does not imply independence

ProofFoundation⏱ 15 minProbabilityindependencecounterexample

Theory: Probability Spaces and Random Variables →

Problem

Let $X_1,X_2$ be independent uniform signs on $\{-1,+1\}$ and $X_3 = X_1X_2$ . Show that $X_1,X_2,X_3$ are pairwise independent but not independent. Then show that Chebyshev's bound on $\Var(X_1+X_2+X_3)$ still applies, while the Chernoff argument does not.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

Hints — 0 of 2 revealed

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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