`r_(xy)lt0`, according as

Options

A. `sigma_(x+y)gtsigma_(x-y)`
B. `sigma_(x+y)ltsigma_(x-y)`
C. `sigma_(x+y)gt0`
D. none of these

Answer

To solve the question regarding the correlation coefficient \( r_{xy} < 0 \), we need to analyze the given options and determine which one corresponds to a negative correlation. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Correlation Coefficient
The correlation coefficient \( r_{xy} \) indicates the strength and direction of a linear relationship between two variables \( x \) and \( y \). If \( r_{xy} < 0 \), it indicates a negative correlation, meaning that as one variable increases, the other tends to decrease.

### Step 2: Analyzing the Options
We have four options to consider:
1. \( \sigma x + y > \sigma x - y \)
2. \( \sigma x + y < \sigma x - y \)
3. \( \sigma x + y > 0 \)
4. None of the above

### Step 3: Using the Formula for Correlation Coefficient
The formula for the correlation coefficient can be expressed as:
\[ r = \frac{\sigma^2(x + y) - \sigma^2(x - y)}{\sigma^2(x + y)} \]
For \( r \) to be negative, the numerator must be less than zero:
\[ \sigma^2(x + y) - \sigma^2(x - y) < 0 \]
This implies:
\[ \sigma^2(x - y) > \sigma^2(x + y) \]

### Step 4: Rearranging the Inequality
From the inequality \( \sigma^2(x - y) > \sigma^2(x + y) \), we can deduce that:
\[ \sigma(x - y) > \sigma(x + y) \]
This means that the sum of the squares of \( x - y \) is greater than the sum of the squares of \( x + y \).

### Step 5: Evaluating the Options
Now, we can evaluate the options based on our findings:
- **Option 1:** \( \sigma x + y > \sigma x - y \) does not necessarily imply a negative correlation.
- **Option 2:** \( \sigma x + y < \sigma x - y \) aligns with our conclusion that \( \sigma^2(x - y) > \sigma^2(x + y) \).
- **Option 3:** \( \sigma x + y > 0 \) does not provide any information about the correlation being negative.
- **Option 4:** None of the above is not applicable since we found a matching option.

### Conclusion
The correct answer is **Option 2: \( \sigma x + y < \sigma x - y \)**.

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