Two variates, x and y, are uncorrelated and have standard deviations,`sigma_(x)` and `sigma_(y)` , respectively. What is the correlation coefficient between x+y and x-y?

Options

A. `(sigma_(x)sigma_(y))/(sigma_(x)^(2)+sigma_(y)^(2))`
B. `(sigma_(x)+sigma_(y))/(2sigma_(x)sigma_(y))`
C. `(sigma^(2)-sigma^(2))/(sigma_(x)^(2)+sigma_(y)^(2))`
D. `(sigma_(y)-sigma_(y))/(sigma_(x)sigma_(y))`

Answer

To find the correlation coefficient between \(x+y\) and \(x-y\), we start by using the properties of correlation and variance.

### Step-by-Step Solution:

1. **Understand the correlation coefficient formula**:
The correlation coefficient \( r \) between two variables \( A \) and \( B \) is given by:
\[ r(A, B) = \frac{\text{Cov}(A, B)}{\sigma_A \sigma_B} \]
where \( \text{Cov}(A, B) \) is the covariance between \( A \) and \( B \), and \( \sigma_A \) and \( \sigma_B \) are the standard deviations of \( A \) and \( B \), respectively.

2. **Define \( A \) and \( B \)**:
Let \( A = x + y \) and \( B = x - y \).

3. **Calculate the covariance \( \text{Cov}(A, B) \)**:
Using the property of covariance:
\[ \text{Cov}(A, B) = \text{Cov}(x+y, x-y) = \text{Cov}(x, x) + \text{Cov}(x, -y) + \text{Cov}(y, x) + \text{Cov}(y, -y) \]
Since \( \text{Cov}(x, x) = \sigma_x^2 \), \( \text{Cov}(y, y) = \sigma_y^2 \), and \( \text{Cov}(x, y) = 0 \) (because \( x \) and \( y \) are uncorrelated), we have:
\[ \text{Cov}(A, B) = \sigma_x^2 - 0 + 0 - \sigma_y^2 = \sigma_x^2 - \sigma_y^2 \]

4. **Calculate the variances \( \sigma_A^2 \) and \( \sigma_B^2 \)**:
- For \( A = x + y \):
\[ \sigma_A^2 = \text{Var}(x+y) = \text{Var}(x) + \text{Var}(y) + 2\text{Cov}(x, y) = \sigma_x^2 + \sigma_y^2 + 0 = \sigma_x^2 + \sigma_y^2 \]
- For \( B = x - y \):
\[ \sigma_B^2 = \text{Var}(x-y) = \text{Var}(x) + \text{Var}(y) - 2\text{Cov}(x, y) = \sigma_x^2 + \sigma_y^2 - 0 = \sigma_x^2 + \sigma_y^2 \]

5. **Calculate the standard deviations**:
Since \( \sigma_A^2 = \sigma_x^2 + \sigma_y^2 \) and \( \sigma_B^2 = \sigma_x^2 + \sigma_y^2 \), we have:
\[ \sigma_A = \sqrt{\sigma_x^2 + \sigma_y^2}, \quad \sigma_B = \sqrt{\sigma_x^2 + \sigma_y^2} \]

6. **Substitute values into the correlation coefficient formula**:
Now, substituting into the correlation coefficient formula:
\[ r(A, B) = \frac{\sigma_x^2 - \sigma_y^2}{\sqrt{\sigma_x^2 + \sigma_y^2} \cdot \sqrt{\sigma_x^2 + \sigma_y^2}} = \frac{\sigma_x^2 - \sigma_y^2}{\sigma_x^2 + \sigma_y^2} \]

### Final Result:
The correlation coefficient between \(x+y\) and \(x-y\) is:
\[ r(x+y, x-y) = \frac{\sigma_x^2 - \sigma_y^2}{\sigma_x^2 + \sigma_y^2} \]