UPSC NDA I 2019 Mathematics Solved Paper

Concept:The correlation coefficient is unaffected by adding a constant to a variable or scaling it by a positive constant.Explanation:Given: correlation between $X$ and $Y$ is $\rho_{XY} = 0.6$.Define $Z = X + 5$ and $W = \frac{Y}{3}$.Step 1: Variance and standard deviation of $Z$ and $W$.Adding a constant does not change variance: $\operatorname{Var}(Z) = \operatorname{Var}(X)$, so $\sigma_Z = \sigma_X$.Scaling multiplies variance by the square of the factor: $\operatorname{Var}(W) = \frac{1}{9}\operatorname{Var}(Y)$, so $\sigma_W = \frac{\sigma_Y}{3}$.Step 2: Covariance between $Z$ and $W$.$\operatorname{Cov}(Z,W) = \operatorname{Cov}\left(X+5,\; \frac{Y}{3}\right) = \frac{1}{3}\operatorname{Cov}(X,Y)$ because adding a constant does not affect covariance and scaling one variable scales the covariance by that factor.Step 3: Correlation coefficient formula.$\rho_{ZW} = \frac{\operatorname{Cov}(Z,W)}{\sigma_Z \sigma_W} = \frac{\frac{1}{3}\operatorname{Cov}(X,Y)}{\sigma_X \cdot \frac{\sigma_Y}{3}} = \frac{\operatorname{Cov}(X,Y)}{\sigma_X \sigma_Y} = \rho_{XY} = 0.6$.Thus, the correlation between $Z$ and $W$ equals the original correlation of $X$ and $Y$.Answer:$0.6$, which corresponds to option D.