Question

Let a random variable X has mean $\mu_x$ and non-zero variance $\sigma ^2 _x$, and another X random variable Y has mean $\mu_y$ and non zero variance $\sigma ^2 _y$. If the correlation Y coefficient between X and Y is $\rho$, then which of the following is/are CORRECT?

• A. $|\rho| \leq 1$
• B. The regression line of Y on X is $y = \mu_y + \frac{\rho \sigma_x}{ \sigma_y} (x − \mu_x )$
• C. The variance of X − Y is $\sigma^2 _x + \sigma^2 _y − 2\rho \sigma_x \sigma_y$
• D. $\rho = 0$ implies X and Y are independent random variables

Answer 1

The Correct Option is A, C

Analysis of the Given Statements on Correlation and Variance

Option (A): |ρ| ≤ 1 

• Correct. The correlation coefficient ρ measures the strength and direction of the linear relationship between X and Y.
• By definition, it is bounded between -1 and 1, i.e., -1 ≤ ρ ≤ 1.
• Thus, |ρ| ≤ 1 is always true.

Option (B): Regression Line of Y on X

• Incorrect. The given regression equation is:
• The correct formula for the regression line of Y on X is:
• The coefficient of (x − μ_X) should be ρσ_Y / σ_X, not ρσ_X / σ_Y.
• Thus, the given equation is incorrect.

Option (C): Variance of (X − Y)

• Correct. The variance of X - Y is given by:
• Since Cov(X, Y) = ρσ_Xσ_Y, we substitute:
• Thus, this statement is correct.

Option (D): ρ = 0 Implies Independence

• Incorrect. While ρ = 0 means that X and Y are uncorrelated, it does not necessarily imply that they are independent.
• Uncorrelated variables can still have a non-linear relationship and may not be completely independent.
• True independence requires that P(X, Y) = P(X)P(Y) for all values of X and Y, which is a stronger condition than zero correlation.

Final Answer

The correct options are:

• (A) |ρ| ≤ 1
• (C) Variance formula: σ_X² + σ_Y² − 2ρσ_Xσ_Y