In a bivariate distribution, it was found `sigma_(x)=3`, the regression line of Y on X is `8x-10y+66=0` while regression line of X on Y is `40x-18y -214=0`. Calculate r(x,y)

Answer

To solve the problem, we need to calculate the correlation coefficient \( r(x,y) \) using the given regression lines and the standard deviation of \( x \).

### Step-by-Step Solution:

1. **Identify the Regression Equations:**
- The regression line of \( Y \) on \( X \) is given by:
\[ 8x - 10y + 66 = 0 \]
- The regression line of \( X \) on \( Y \) is given by:
\[ 40x - 18y - 214 = 0 \]

2. **Rearranging the Regression Line of \( Y \) on \( X \):**
- Rearranging the first equation to express \( y \) in terms of \( x \):
\[ 10y = 8x + 66 \implies y = \frac{8}{10}x + \frac{66}{10} \implies y = \frac{4}{5}x + 6.6 \]
- The slope (regression coefficient) \( b_{yx} = \frac{4}{5} \).

3. **Rearranging the Regression Line of \( X \) on \( Y \):**
- Rearranging the second equation to express \( x \) in terms of \( y \):
\[ 40x = 18y + 214 \implies x = \frac{18}{40}y + \frac{214}{40} \implies x = \frac{9}{20}y + 5.35 \]
- The slope (regression coefficient) \( b_{xy} = \frac{9}{20} \).

4. **Using the Formula for Correlation Coefficient:**
- The correlation coefficient \( r(x,y) \) can be calculated using the relationship:
\[ r(x,y) = \sqrt{b_{yx} \cdot b_{xy}} \]
- Substituting the values of \( b_{yx} \) and \( b_{xy} \):
\[ r(x,y) = \sqrt{\left(\frac{4}{5}\right) \cdot \left(\frac{9}{20}\right)} \]

5. **Calculating the Product:**
- First, calculate the product:
\[ \frac{4}{5} \cdot \frac{9}{20} = \frac{36}{100} = \frac{9}{25} \]

6. **Taking the Square Root:**
- Now, take the square root:
\[ r(x,y) = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

7. **Determining the Sign of \( r(x,y) \):**
- Since both regression coefficients \( b_{yx} \) and \( b_{xy} \) are positive, the correlation coefficient \( r(x,y) \) is also positive.

### Final Result:
\[ r(x,y) = \frac{3}{5} \]