Question

Let \( \Theta \) be a random variable having \( U(0, 2\pi) \) distribution. Let \( X = \cos \Theta \) and \( Y = \sin \Theta \). Let \( \rho \) be the correlation coefficient between \( X \) and \( Y \). Then \( 100\rho \) is equal to __________ (answer in integer).

Answer 1

Correct Answer: 0

1. Independence of \( X \) and \( Y \): - Since \( \Theta \sim U(0, 2\pi) \), the random variables \( \cos\Theta \) and \( \sin\Theta \) are uncorrelated, as: \[ E(\cos\Theta \cdot \sin\Theta) = 0. \] - Additionally, the joint distribution of \( X \) and \( Y \) is symmetric over the unit circle, leading to zero covariance: \[ \text{Cov}(X, Y) = E(XY) - E(X)E(Y) = 0. \] 2. Correlation Coefficient: - The correlation coefficient \( \rho \) is defined as: \[ \rho = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}. \] - Since \( \text{Cov}(X, Y) = 0 \), it follows that: \[ \rho = 0. \] 3. Value of \( 100\rho \): - Multiply \( \rho \) by 100: \[ 100\rho = 100 \cdot 0 = 0. \]