Which of the following statements about covariance is correct?

• A. A covariance of $0$ always means the variables are independent.
• B. Covariance is unaffected by the scale of the variables.
• C. Covariance measures the direction of the linear relationship between two random variables.
• D. Covariance is always a value between $0$ and $1$.

Answer: (C)

Verified step by step guidance

1. Recall the definition of covariance: for two random variables X and Y, covariance is given by the formula \(\text{Cov}(X,Y) = E[(X - E[X])(Y - E[Y])]\), where \(E\) denotes the expected value.
2. Understand that covariance measures how two variables change together. A positive covariance indicates that the variables tend to increase or decrease together, while a negative covariance indicates that one tends to increase when the other decreases.
3. Note that a covariance of zero means there is no linear relationship between the variables, but it does not necessarily imply independence, because variables can be dependent in a nonlinear way.
4. Recognize that covariance is affected by the scale of the variables. If you multiply a variable by a constant, the covariance changes accordingly, so it is not scale-invariant.
5. Remember that covariance is not restricted to values between 0 and 1; it can take any real value depending on the variables' distributions and scales.