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Answer: If two variables have a correlation of zero, this implies they also have zero dependence between them
**Explanation:** Statement D is false because correlation measures only **linear** relationships between variables. A correlation of zero indicates no linear relationship, but there could still be **non-linear** dependencies between the variables. **Key points:** - **Correlation (ρ)** = Cov(X,Y) / (σ_X × σ_Y) - **Correlation = 0** means no linear relationship - **Dependence ≠ 0** could still exist through non-linear relationships - **Independence** implies correlation = 0, but correlation = 0 does NOT imply independence **Examples of zero correlation but non-zero dependence:** 1. X ~ Uniform(-1,1), Y = X² → Correlation = 0 but clear quadratic relationship 2. Circular relationships where points form a circle pattern 3. Other non-linear functional relationships **Correct statements:** - A: True - Both measure linear relationship strength - B: True - This is the mathematical definition of correlation - C: True - This is the definition of statistical independence
Author: Nikitesh Somanthe
Which of the following statements is false?
A
Correlation and covariance measure the strength of the linear relationship between two variables
B
Mathematically, we determine correlation by dividing the covariance between two random variables by the product of their standard deviations
C
Two variables are independent if the knowledge of one variable does not impact the probability distribution of the other variable
D
If two variables have a correlation of zero, this implies they also have zero dependence between them
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