The equations of two lines of regression are `3x+2y-26=0` and `6x +y- 31=0`. find variance of `x` if variance of `y` is 36.

Answer

To find the variance of \( x \) given the equations of two lines of regression and the variance of \( y \), we can follow these steps:

### Step 1: Identify the regression equations
The given regression equations are:
1. \( 3x + 2y - 26 = 0 \) (assumed to be the regression of \( y \) on \( x \))
2. \( 6x + y - 31 = 0 \) (assumed to be the regression of \( x \) on \( y \))

### Step 2: Convert the regression equations to slope-intercept form
For the first equation:
\[ 2y = -3x + 26 \implies y = -\frac{3}{2}x + 13 \]
This gives us the regression coefficient \( b_{yx} = -\frac{3}{2} \).

For the second equation:
\[ y = -6x + 31 \implies 6x = -y + 31 \implies x = -\frac{1}{6}y + \frac{31}{6} \]
This gives us the regression coefficient \( b_{xy} = -\frac{1}{6} \).

### Step 3: Calculate the correlation coefficient \( R \)
The correlation coefficient \( R \) can be calculated using the formula:
\[ R^2 = b_{xy} \cdot b_{yx} \]
Substituting the values:
\[ R^2 = \left(-\frac{1}{6}\right) \cdot \left(-\frac{3}{2}\right) = \frac{3}{12} = \frac{1}{4} \]
Thus, \( R = -\frac{1}{2} \) (since both coefficients are negative, \( R \) is negative).

### Step 4: Use the variance of \( y \) to find the variance of \( x \)
We know that the variance of \( y \) is given as \( \sigma_y^2 = 36 \). Therefore, the standard deviation \( \sigma_y \) is:
\[ \sigma_y = \sqrt{36} = 6 \]

Using the relationship between the variances:
\[ b_{yx} = R \cdot \frac{\sigma_y}{\sigma_x} \]
We can rearrange this to find \( \sigma_x \):
\[ \sigma_x = R \cdot \frac{\sigma_y}{b_{yx}} \]
Substituting the known values:
\[ \sigma_x = -\frac{1}{2} \cdot \frac{6}{-\frac{3}{2}} = -\frac{1}{2} \cdot \frac{6 \cdot 2}{-3} = \frac{1 \cdot 6 \cdot 2}{3} = 4 \]

### Step 5: Calculate the variance of \( x \)
The variance of \( x \) is given by:
\[ \sigma_x^2 = \sigma_x^2 = 4^2 = 16 \]

### Final Answer
The variance of \( x \) is \( 16 \).

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