If the Pearson correlation coefficient between two random variables $X$ and $Y$ is zero ($r_{xy} = 0$), it implies that:
Options:
(A) There is no linear relationship between $X$ and $Y$
(B) $X$ and $Y$ are strictly statistically independent in all cases
(C) $Y$ is a deterministic quadratic function of $X$
(D) The variance of $X$ equals the variance of $Y$
Answer Key:
A
Step-by-Step Solution & Explanation
Pearson's correlation coefficient measures the degree of linear association between two variables. When $r = 0$, there is no linear association. However, $X$ and $Y$ might still have a strong non-linear relationship (such as $Y = X^2$ where $r = 0$ but $Y$ is fully determined by $X$). Thus, $r = 0$ implies no linear relationship.