If the regression equation of y is `y=lambdax+4` and that of X on Y be 4x=y-5 then

Options

A. `0 lt 4 lambda le 1`
B. `0 le lambda le 4`
C. `0 lt lambda le 4`
D. None of these

Answer

To solve the given problem, we will analyze the regression equations provided and derive the necessary conditions for the parameter \( \lambda \).

### Step-by-Step Solution:

1. **Identify the Given Regression Equations**:
- The regression equation of \( y \) on \( x \) is given as:
\[ y = \lambda x + 4 \]
- The regression equation of \( x \) on \( y \) is given as:
\[ 4x = y - 5 \]

2. **Rearranging the Second Equation**:
- We can rearrange the second equation to express \( x \) in terms of \( y \):
\[ 4x = y - 5 \implies x = \frac{y - 5}{4} \]

3. **Finding the Slope of the Regression**:
- The slope of the regression line of \( y \) on \( x \) is represented as \( b_{y|x} = \lambda \).
- The slope of the regression line of \( x \) on \( y \) can be calculated as follows:
- From \( x = \frac{y - 5}{4} \), we can express it in the form \( x = \frac{1}{4}y - \frac{5}{4} \).
- Thus, the slope \( b_{x|y} = \frac{1}{4} \).

4. **Using the Relationship Between Slopes**:
- The relationship between the slopes of the regression lines is given by:
\[ b_{y|x} \cdot b_{x|y} = r^2 \]
- Here, \( r \) is the correlation coefficient. Since both regression slopes must be positive, we have:
\[ \lambda \cdot \frac{1}{4} = r^2 \]

5. **Finding the Range of \( \lambda \)**:
- Since \( r^2 \) must be between 0 and 1 (i.e., \( 0 \leq r^2 \leq 1 \)), we can derive:
\[ 0 \leq \lambda \cdot \frac{1}{4} \leq 1 \]
- Multiplying through by 4 gives:
\[ 0 \leq \lambda \leq 4 \]

6. **Conclusion**:
- Therefore, the value of \( \lambda \) must satisfy:
\[ 0 < \lambda \leq 4 \]

### Final Answer:
The solution concludes that:
\[ 0 < \lambda \leq 4 \]