State true or false: Both the regression coefficients cannot exceed 1

Answer

To determine whether the statement "Both the regression coefficients cannot exceed 1" is true or false, we will analyze the properties of regression coefficients and their relationship with the correlation coefficient.

### Step-by-Step Solution:

1. **Understanding Regression Coefficients**:
- The regression coefficient of X on Y is denoted as \( B_{XY} \).
- The regression coefficient of Y on X is denoted as \( B_{YX} \).

2. **Correlation Coefficient**:
- The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. It has a range of \( -1 \) to \( 1 \).

3. **Relationship Between Regression Coefficients and Correlation Coefficient**:
- The relationship between the regression coefficients and the correlation coefficient is given by the formula:
\[ r^2 = B_{XY} \times B_{YX} \]
- Since \( r^2 \) is the square of the correlation coefficient, it lies between \( 0 \) and \( 1 \) (i.e., \( 0 \leq r^2 \leq 1 \)).

4. **Implication of the Relationship**:
- From the equation \( r^2 = B_{XY} \times B_{YX} \), we can infer that:
\[ 0 \leq B_{XY} \times B_{YX} \leq 1 \]
- This means that the product of the two regression coefficients is always less than or equal to 1.

5. **Analyzing the Product**:
- For the product \( B_{XY} \times B_{YX} \) to be less than or equal to 1, at least one of the coefficients must be less than or equal to 1.
- If both \( B_{XY} \) and \( B_{YX} \) were greater than 1, their product would exceed 1, which contradicts our earlier finding.

6. **Conclusion**:
- Therefore, it is true that both regression coefficients cannot exceed 1. If one coefficient were to exceed 1, the other must be less than 1 to keep the product within the bounds of 1.

### Final Answer:
The statement "Both the regression coefficients cannot exceed 1" is **True**.