If one of the regression coefficient is unity, the other must be ……….. Unity.

Answer

To solve the question, we need to understand the relationship between regression coefficients and the correlation coefficient. Here’s a step-by-step breakdown:

### Step 1: Understanding Regression Coefficients
Regression coefficients are used to describe the relationship between two variables in a regression analysis. If we denote the regression coefficients as \( b_1 \) and \( b_2 \), they represent the slopes of the regression lines.

### Step 2: Correlation Coefficient
The correlation coefficient \( r \) is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1.

### Step 3: Relationship Between Regression Coefficients and Correlation Coefficient
The correlation coefficient \( r \) can be expressed in terms of the regression coefficients:
\[ r = \sqrt{b_1 \cdot b_2} \]
where \( b_1 \) is the regression coefficient of Y on X and \( b_2 \) is the regression coefficient of X on Y.

### Step 4: Analyzing the Given Condition
In the question, we are given that one of the regression coefficients is unity (i.e., \( b_1 = 1 \)). We need to find out what the other regression coefficient \( b_2 \) must be.

### Step 5: Substituting into the Formula
Substituting \( b_1 = 1 \) into the correlation coefficient formula:
\[ r = \sqrt{1 \cdot b_2} = \sqrt{b_2} \]
Since \( r \) must lie between -1 and 1, \( \sqrt{b_2} \) must also lie within this range.

### Step 6: Finding the Value of \( b_2 \)
For \( r \) to be valid, \( b_2 \) must be less than or equal to 1 (since \( \sqrt{b_2} \) cannot exceed 1). Thus, we conclude:
\[ b_2 \leq 1 \]
This means that if one regression coefficient is unity, the other must be less than or equal to unity.

### Conclusion
Therefore, the answer to the question is:
If one of the regression coefficients is unity, the other must be less than or equal to unity.

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