Question

Let \( X_1 \) and \( X_2 \) be independent \( N(0,1) \) random variables. Define \[ \text{sgn}(u) = \begin{cases} -1, & \text{if } u < 0 \\ 0, & \text{if } u = 0 \\ 1, & \text{if } u > 0 \end{cases} \] Let \( Y_1 = X_1 \, \text{sgn}(X_2) \) and \( Y_2 = X_2 \, \text{sgn}(X_1) \). If the correlation coefficient between \(Y_1\) and \(Y_2\) is \(\alpha\), then \(\pi \alpha\) is equal to ............

Hint:

For symmetric normal variables, use quadrant symmetry. The correlation between sign-modified Gaussian pairs often leads to expressions involving \(\frac{2}{\pi}\).

Answer 1

Correct Answer: 2

Step 1: Express correlation.
\[ \alpha = \frac{\text{Cov}(Y_1, Y_2)}{\sqrt{\text{Var}(Y_1) \, \text{Var}(Y_2)}}. \] Since \(Y_1, Y_2\) have same distribution as \(X_1, X_2\), \(\text{Var}(Y_1) = \text{Var}(Y_2) = 1.\)
Step 2: Compute covariance.
\[ \text{Cov}(Y_1, Y_2) = E[Y_1 Y_2] = E[X_1 X_2 \, \text{sgn}(X_1 X_2)]. \] Since \(\text{sgn}(X_1 X_2) = 1\) if \(X_1 X_2>0\) and \(-1\) otherwise, \[ E[Y_1 Y_2] = E[|X_1 X_2|] - E[-|X_1 X_2|] = 2E[|X_1 X_2| \, I(X_1 X_2>0)] - E[|X_1 X_2|]. \]
Step 3: Simplify using symmetry.
Since \(X_1, X_2\) are independent and symmetric, \[ E[Y_1 Y_2] = \frac{2}{\pi}. \]
Step 4: Compute \(\pi \alpha.\)
\[ \alpha = \frac{2}{\pi} \Rightarrow \pi \alpha = 2. \] Final Answer: \[ \boxed{2} \]