# If you regress random variable Y against random variable X, would the results be the same if you regressed X against Y?

They would be the same only if the

The Pearson correlation coefficient will always be the same regardless what the variances are. However, the regression lines will have different slopes (hence, not the same) unless the variances are equal.

This may be more technical than what you are looking for, but here is the slope of the regression line for both cases:

As you can see, they will only have the same slope if the variances are equal.

hope that helped

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No, the results would not necessarily be the same if you regress random variable Y against random variable X compared to if you regress X against Y. The regression results may differ in terms of the coefficients, the intercept, and the goodness of fit statistics such as R-squared. This is because regression analysis estimates the relationship between variables based on the direction of causality assumed or implied. When regressing Y against X, the model aims to explain the variation in Y using X as the predictor variable. Conversely, when regressing X against Y, the model aims to explain the variation in X using Y as the predictor variable. Therefore, the relationships and strengths of association between the variables may vary depending on which variable is considered the predictor and which one is considered the response.

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