Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Independent Random Variables
Let (X1, Y1), (X2, Y2) and (X3, Y3) be independent and identically distributed (i.i.d.) random vectors following a bivariate normal distribution with mean vector (0, 0)
and correlation matrix \(\left(\begin{array}{ll} 1 & \rho \\ \rho & 1 \end{array}\right)\), where |ρ| < 1. Suppose that
Sρ = 3 E(sgn (X1 - X2)(Y1 - Y3)),
where
\(\operatorname{sgn}(x)=\left\{\begin{array}{cc} \frac{x}{|x|}, & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array} .\right.\)
Then which of the following statements are true?
1
If X1 and Y1 are independent random variables, then Sρ = 0
2
\(S_\rho=\frac{6}{\pi} \sin ^{-1} \frac{\rho}{2}\)
3
If Sρ = 0, then X1 and Y1 are independent random variables
4
If X1 and Y1 are independent random variables, then Sρ = \(\frac{1}{2}\)