Arithmetic mean of positive values of regression coefficients is greater than or equal to ______

Answer

To solve the problem, we need to find the relationship between the arithmetic mean of the positive values of regression coefficients and a certain number. Here’s a step-by-step solution:

### Step 1: Define the Regression Coefficients
Let:
- \( B_{Y,X} \) = Regression coefficient of Y on X
- \( B_{X,Y} \) = Regression coefficient of X on Y

### Step 2: Understand the Relationship with Correlation Coefficient
The correlation coefficient \( r \) is related to the regression coefficients by the formula:
\[ r^2 = B_{X,Y} \cdot B_{Y,X} \]
Since we are dealing with positive values of regression coefficients, we can take the square root:
\[ r = \sqrt{B_{X,Y} \cdot B_{Y,X}} \]

### Step 3: Apply the Arithmetic Mean-Geometric Mean Inequality
For any two positive numbers \( a \) and \( b \), the Arithmetic Mean-Geometric Mean (AM-GM) inequality states:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
In our case, we can apply this to the regression coefficients:
\[ \frac{B_{Y,X} + B_{X,Y}}{2} \geq \sqrt{B_{Y,X} \cdot B_{X,Y}} \]

### Step 4: Relate to the Correlation Coefficient
From the previous step, we know that:
\[ \sqrt{B_{Y,X} \cdot B_{X,Y}} = r \]
Thus, we can rewrite the AM-GM inequality as:
\[ \frac{B_{Y,X} + B_{X,Y}}{2} \geq r \]

### Conclusion
Therefore, the arithmetic mean of the positive values of regression coefficients \( B_{Y,X} \) and \( B_{X,Y} \) is greater than or equal to the correlation coefficient \( r \).

The final answer is:
**The arithmetic mean of positive values of regression coefficients is greater than or equal to \( r \).**

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