Part CCSIR NET December 2023spearman-for-bivariate-normal

Spearman for bivariate normal

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), (X_{3}, Y_{3})$ be i.i.d. bivariate normal with mean (0, 0) and correlation matrix $[[1, ρ], [ρ, 1]]$ , | $ρ$ | < 1. Let $S_{ρ} = 3 E ($ sgn $(X_{1} - X_{2}) (Y_{1} - Y_{3}))$ . Which of the following statements are true?

A.If $X_{1}$ and $Y_{1}$ are independent, then $S_{ρ} = 0$
B. $S_{ρ} = (6/ π) sin$ ⁻ $^{1} (ρ /2)$
C.If $S_{ρ} = 0$ , then $X_{1}$ and $Y_{1}$ are independent
D.If $X_{1}$ and $Y_{1}$ are independent, then $S_{ρ} = 1/2$

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Spearman for bivariate normal

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