Angle between the two lines of regression is given by

Options

A. `tan^(-1){(b_(YX)-(1)/(b_(XY)))/(1+(b_(XY))/(b_(YX)))}`
B. `tan^(-1){(b_(YX)b_(XY)-1)/(b_(YX)+b_(XY))}`
C. `tan^(-1){(b_(YX)-b_(XY))/(1+b_(YX).b_(XY))}`
D. none of these

Answer

To find the angle between the two lines of regression, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Regression Lines**:
The two lines of regression can be represented as:
- Regression of Y on X: \( y = b_{yx} \cdot x + c_1 \)
- Regression of X on Y: \( x = b_{xy} \cdot y + c_2 \)

2. **Determine the Slopes**:
- The slope of the first regression line (Y on X) is \( m_1 = b_{yx} \).
- Rearranging the second regression line gives us:
\[ x - c_2 = b_{xy} \cdot y \implies y = \frac{x - c_2}{b_{xy}} \implies y = \frac{1}{b_{xy}} \cdot x - \frac{c_2}{b_{xy}} \]
Hence, the slope of the second regression line (X on Y) is \( m_2 = \frac{1}{b_{xy}} \).

3. **Use the Formula for Tangent of the Angle**:
The formula for the tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 \cdot m_2} \]

4. **Substitute the Slopes**:
Substituting \( m_1 \) and \( m_2 \) into the formula:
\[ \tan \theta = \frac{b_{yx} - \frac{1}{b_{xy}}}{1 + b_{yx} \cdot \frac{1}{b_{xy}}} \]

5. **Simplify the Expression**:
To simplify:
\[ \tan \theta = \frac{b_{yx} \cdot b_{xy} - 1}{b_{xy} + b_{yx}} \]

6. **Find the Angle**:
To find \( \theta \), we take the inverse tangent:
\[ \theta = \tan^{-1}\left(\frac{b_{yx} \cdot b_{xy} - 1}{b_{xy} + b_{yx}}\right) \]

7. **Match with Given Options**:
The final expression can be matched with the options provided in the question. The correct option that matches our derived expression is:
\[ \tan^{-1}\left(\frac{b_{yx} - b_{xy}}{1 + b_{yx} \cdot b_{xy}}\right) \]

### Conclusion:
Thus, the angle between the two lines of regression is given by:
\[ \theta = \tan^{-1}\left(\frac{b_{yx} - b_{xy}}{1 + b_{yx} \cdot b_{xy}}\right) \]