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Exercise 2.24 (Independence and orthogonality)

Let $X$ be a random variable taking values in $R^{n}$ with the Gaussian distribution, and let $v_{1}, \dots, v_{m}$ be vectors in $R^{n}$ . Show that the random variables $X \cdot v_{1}, \dots, X \cdot v_{m}$ (with $⋅$ denoting the Euclidean inner product) are jointly independent if and only if the $v_{1}, \dots, v_{m}$ are pairwise orthogonal.

Exercise 2.24 (Independence and orthogonality)

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