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 problem, we need to analyze the given regression equations and determine the possible values for the parameter \( \lambda \).

### Step 1: Write down the regression equations
We have two regression equations:
1. \( y = \lambda x + 4 \) (Regression of \( y \) on \( x \))
2. \( 4x = y - 5 \) (Regression of \( x \) on \( y \))

### Step 2: Rearranging the second equation
We can rearrange the second equation to express \( x \) in terms of \( y \):
\[ 4x = y - 5 \implies x = \frac{1}{4}y - \frac{5}{4} \]

### Step 3: Identify the regression coefficients
From the equations, we can identify the regression coefficients:
- For the regression of \( y \) on \( x \), the coefficient \( b_{xy} = \lambda \).
- For the regression of \( x \) on \( y \), we can see that the coefficient \( b_{yx} = \frac{1}{4} \).

### Step 4: Use the properties of regression coefficients
We know two important properties of regression coefficients:
1. If one regression coefficient is positive, the other must also be positive.
2. If one coefficient is less than unity, the other must be greater than unity.

From our equations:
- Since \( b_{yx} = \frac{1}{4} < 1 \), it follows that \( b_{xy} = \lambda > 1 \).

### Step 5: Determine the range of \( \lambda \)
From the properties we established:
- Since \( \lambda > 1 \), we can conclude that \( \lambda \) must be greater than 1.

### Step 6: Conclusion
Thus, the range of \( \lambda \) is:
\[ \lambda > 1 \]