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?

Updated On: Feb 10, 2025
  • $|\rho| \leq 1$
  • The regression line of Y on X is $y = \mu_y + \frac{\rho \sigma_x}{ \sigma_y} (x βˆ’ \mu_x )$
  • The variance of X βˆ’ Y is $\sigma^2 _x + \sigma^2 _y βˆ’ 2\rho \sigma_x \sigma_y $
  • $\rho = 0$ implies X and Y are independent random variables
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The Correct Option is A, C

Solution and Explanation

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 βˆ’ ΞΌX) should be ρσY / ΟƒX, not ρσX / ΟƒY.
  • Thus, the given equation is incorrect.

Option (C): Variance of (X βˆ’ Y)

  • Correct. The variance of X - Y is given by:
  • Since Cov(X, Y) = ρσXΟƒY, 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: ΟƒX2 + ΟƒY2 βˆ’ 2ρσXΟƒY
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