4 For real numbers and , let sgn() 1 if 0 0 if 0 1 if 0 Prove that for random variables X and Y , (1X 2, 1Y 2) (X, Y ) sgn(11)

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Question: 4. For real numbers and , let sgn() = 1 if > 0 0 if = 0 1 if

4. For real numbers α and β, let sgn(αβ) =    1 if αβ > 0 0 if αβ = 0 −1 if αβ < 0. Prove that for random variables X and Y , ρ(α1X + α2, β1Y + β2) = ρ(X, Y ) sgn(α1β1).

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