Which of the following is not possible

Options

A. `byx gt 0 and bxy lt0`
B. `byx. Bxy gt 1`
C. `{e (x,y)}^(2) le 1`
D. None of these

Answer

To solve the question, we need to determine which of the given options regarding correlation and its properties is not possible. Let's break down the solution step by step.

### Step 1: Understanding Correlation Coefficient
The correlation coefficient (denoted as \( r \)) measures the strength and direction of a linear relationship between two variables. The value of \( r \) lies between -1 and 1, inclusive. Therefore, we have:
\[ -1 \leq r \leq 1 \]

### Step 2: Analyzing the Options
We need to analyze the statements provided in the options to see which one contradicts the properties of the correlation coefficient.

1. **Option 1**: If one variable is positive, the other must be negative.
- This statement contradicts the property of correlation. If one variable is positive, the correlation does not imply that the other variable must be negative. Therefore, this option is **not possible**.

2. **Option 2**: The product of the correlation coefficients \( b_{yx} \) and \( b_{xy} \) is greater than 1.
- The product of the correlation coefficients must also lie within the bounds defined by the correlation properties. Since both coefficients are between -1 and 1, their product cannot exceed 1. Hence, this statement is also **not possible**.

### Step 3: Conclusion
Based on the analysis:
- Both **Option 1** and **Option 2** are not possible according to the properties of correlation.

### Final Answer
Thus, the answer to the question "Which of the following is not possible?" is both **Option 1** and **Option 2**.

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