Relationship Between Variables X and Y

Solution & Explanation

1. Regression of Y on X

When we perform a regression analysis of Y on X, we are essentially trying to model Y as a function of X. In the given scenario, the relationship between Y and X is defined by the equation:

$Y = X + \epsilon$

where $\epsilon$ is a random normal noise term. The goal of linear regression is to find the line that best fits the data points, minimizing the sum of squared differences between the observed and predicted values of Y.

• Coefficient (Slope):

  • The equation can be rewritten as: $Y = 1 \cdot X + \epsilon$
  • Here, the coefficient (or slope) of X is 1, indicating that for every unit increase in X, Y increases by one unit. This is because the relationship is linear with a slope of 1.

• Intercept:

  • The intercept is the expected value of Y when X is zero. In this case, the intercept would be the expected value of the noise term $\epsilon$, which is typically zero if the noise is normally distributed with a mean of zero.

• R-squared Value:

  • The R-squared value will be less than 1 because the presence of noise ($\epsilon$) introduces variability in Y that cannot be explained by X alone.

2. Regression of X on Y

In this reversed scenario, we are trying to model X as a function of Y. The relationship can be expressed as:

$X = Y - \epsilon$

• Coefficient (Slope):

  • The coefficient in this case is still 1, as the relationship is symmetric. For every unit change in Y, there is a corresponding unit change in X.

• Intercept:

  • The intercept is the expected value of $-\epsilon$, which is zero if $\epsilon$ has a mean of zero.

• R-squared Value:

  • Similar to the previous scenario, the R-squared value will be less than 1 due to the noise component, which adds unexplained variability.

Conclusion:

In both regression scenarios (Y on X and X on Y), the coefficient (slope) is 1, indicating a unit-to-unit relationship between X and Y. The intercepts are zero, assuming the noise term has a mean of zero. The R-squared values in both cases will be less than 1, reflecting the variability introduced by the random noise term. This demonstrates the symmetric nature of the linear relationship between X and Y when defined simply as a linear function with added noise.