The covariance is not a good measure of the strength of a...

The covariance is not a good measure of the strength of a relationship between two random variables because it depends so much on the units of measurement of the two variables. However, if $X$ and $Y$ are standardized random variables, this problem should be eliminated. Is it?

Key Concepts

Covariance

Covariance is a measure of the degree to which two random variables vary together. Its value depends on the units of the variables, meaning that different scales result in different covariance values, which hampers direct comparison of the strength of relationships across different datasets or different unit systems.

Standardization

Standardization is the process of rescaling random variables so that they have a mean of zero and a variance (or standard deviation) of one. This transformation removes the units from the data, making the measurements dimensionless and thereby allowing for a fairer comparison of relationships between the variables.

Standardized Random Variables

Standardized random variables are those that have been transformed by standardization, resulting in a common scale across variables. With this transformation, the covariance of two standardized variables is equivalent to the correlation coefficient, which is a unitless measure that quantifies the strength and direction of the linear relationship between them.

Correlation

Correlation is a normalized measure of the linear relationship between two variables that has values between -1 and 1. When working with standardized random variables, the covariance directly reduces to the correlation, effectively eliminating the problem of unit dependence and allowing for an assessment of the relationship's strength without the influence of the original measurement scales.

Please give Ace some feedback