If both the regression coefficients `b_(YX) and b_(XY)` are positive, then

Options

A. `(1)/(b_(YX))+(1)/(b_(XY))gt(2)/(r )`
B. `(1)/(b_(YX))+(1)/(b_(XY))lt(2)/(r )`
C. `(1)/(b_(YX))+(1)/(b_(XY))lt(r)/(2)`
D. none of these

Answer

To solve the question, we need to analyze the relationship between the regression coefficients \( b_{YX} \) and \( b_{XY} \) and the correlation coefficient \( r \). Here’s a step-by-step solution:

### Step 1: Understand the Regression Coefficients
The regression coefficients \( b_{YX} \) and \( b_{XY} \) represent the slopes of the regression lines of \( Y \) on \( X \) and \( X \) on \( Y \), respectively.

### Step 2: Relationship Between Regression Coefficients and Correlation Coefficient
The correlation coefficient \( r \) can be expressed in terms of the regression coefficients as follows:
\[ r = \sqrt{b_{YX} \cdot b_{XY}} \]
This indicates that \( r \) is the geometric mean of the two regression coefficients.

### Step 3: Analyze the Given Condition
Given that both \( b_{YX} \) and \( b_{XY} \) are positive, it follows that:
\[ r = \sqrt{b_{YX} \cdot b_{XY}} > 0 \]
This means that the correlation \( r \) is also positive.

### Step 4: Apply the Arithmetic Mean-Geometric Mean Inequality
According to the Arithmetic Mean-Geometric Mean (AM-GM) inequality, for any two positive numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
In our case, we can apply this to \( b_{YX} \) and \( b_{XY} \):
\[ \frac{b_{YX} + b_{XY}}{2} \geq \sqrt{b_{YX} \cdot b_{XY}} = r \]

### Step 5: Conclusion
From the above analysis, we can conclude that if both regression coefficients \( b_{YX} \) and \( b_{XY} \) are positive, then the following inequality holds:
\[ \frac{b_{YX} + b_{XY}}{2} \geq r \]
This indicates that the arithmetic mean of the regression coefficients is greater than or equal to the correlation coefficient.

### Final Answer
Thus, the correct inequality that follows when both regression coefficients are positive is:
\[ \frac{b_{YX} + b_{XY}}{2} \geq r \]

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