Write down the normal equations for the X on Y regression Line

In linear regression, the normal equations are used to find the coefficients of the regression line that minimizes the sum of squared errors between the observed data points and the predicted values on the line. The X on Y regression line, also known as the regression of Y on X, estimates the relationship between the dependent variable Y and the independent variable X. The normal equations for this type of regression line are as follows:

Given a dataset with \(n\) observations:

1. **Regression Line Equation:**
   \[ \hat{Y} = b_0 + b_1X \]

   Where:
   - \(\hat{Y}\) is the predicted value of the dependent variable Y.
   - \(b_0\) is the y-intercept (constant term).
   - \(b_1\) is the slope coefficient (the change in Y for a one-unit change in X).

2. **Normal Equations:**
   To find the coefficients \(b_0\) and \(b_1\), we need to solve the following equations:

   \[ b_1 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^{n} (X_i - \bar{X})^2} \]

   \[ b_0 = \bar{Y} - b_1 \bar{X} \]

   Where:
   - \(X_i\) and \(Y_i\) are the values of the independent and dependent variables for the \(i\)-th observation.
   - \(\bar{X}\) and \(\bar{Y}\) are the means of the independent and dependent variables, respectively.

These equations determine the coefficients \(b_0\) and \(b_1\) that minimize the sum of squared errors between the observed values of Y and the predicted values from the regression line. Once these coefficients are estimated, the regression line equation can be used to predict the value of Y for any given value of X.