Exercise 2.23 (Pairwise independent but not jointly independent random variables)

Question

Give an example of three random variables $X,Y,Z$ which are pairwise independent (that is, any two of $X,Y,Z$ are independent of each other), but not jointly independent.

Elu Thingol · Accepted Answer

\begin{example}
Let ${\bf F}_2$ be the field of two elements, let $V \subset {\bf F}_2^3$ be the subspace of triples $(x_1,x_2,x_3) \in {\bf F}_2^3$ with $x_1+x_2+x_3=0$, and let $(X_1,X_2,X_3)$ be drawn uniformly at random from $V$. Then $(X_1,X_2,X_3)$ are pairwise independent, but not jointly independent. In particular, $X_3$ is independent of each of $X_1,X_2$ separately, but is not independent of $(X_1,X_2)$. For a generalization of this result, see \href{../topics_in_random_matrix_theory/exercise_1-1-12}{this exercise}.
\end{example}

Exercise 2.23 (Pairwise independent but not jointly independent random variables)

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Exercise 2.23 (Pairwise independent but not jointly independent random variables)

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