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_(x)^(2)-sigma_(y)^(2))/(sigma_(x)^(2)+sigma_(y)^(2))`
D. `(sigma_(y)-sigma_(x))/(sigma_(x)sigma_(y))`

Explanation

To find the correlation coefficient between \( x+y \) and \( x-y \), we will follow these steps:

### Step 1: Define the Variates
Let:
- \( u = x + y \)
- \( v = x - y \)

### Step 2: Calculate the Means
The means of \( u \) and \( v \) can be expressed as:
- \( E[u] = E[x+y] = E[x] + E[y] \)
- \( E[v] = E[x-y] = E[x] - E[y] \)

### Step 3: Calculate the Variances
Next, we will calculate the variances of \( u \) and \( v \):
- The variance of \( u \):
\[ \text{Var}(u) = \text{Var}(x+y) = \text{Var}(x) + \text{Var}(y) \quad (\text{since } x \text{ and } y \text{ are uncorrelated}) \]
Let \( \sigma_x \) be the standard deviation of \( x \) and \( \sigma_y \) be the standard deviation of \( y \):
\[ \text{Var}(u) = \sigma_x^2 + \sigma_y^2 \]

- The variance of \( v \):
\[ \text{Var}(v) = \text{Var}(x-y) = \text{Var}(x) + \text{Var}(y) \quad (\text{since } x \text{ and } y \text{ are uncorrelated}) \]
Thus,
\[ \text{Var}(v) = \sigma_x^2 + \sigma_y^2 \]

### Step 4: Calculate the Covariance
Now, we calculate the covariance between \( u \) and \( v \):
\[ \text{Cov}(u, v) = \text{Cov}(x+y, x-y) = \text{Cov}(x, x) + \text{Cov}(x, -y) + \text{Cov}(y, x) + \text{Cov}(y, -y) \]
Since \( \text{Cov}(x, y) = 0 \) (because \( x \) and \( y \) are uncorrelated):
\[ \text{Cov}(u, v) = \text{Var}(x) - \text{Var}(y) = \sigma_x^2 - \sigma_y^2 \]

### Step 5: Calculate the Correlation Coefficient
The correlation coefficient \( \rho \) between \( u \) and \( v \) is given by:
\[ \rho(u, v) = \frac{\text{Cov}(u, v)}{\sqrt{\text{Var}(u) \cdot \text{Var}(v)}} \]
Substituting the values we found:
\[ \rho(u, v) = \frac{\sigma_x^2 - \sigma_y^2}{\sqrt{(\sigma_x^2 + \sigma_y^2)(\sigma_x^2 + \sigma_y^2)}} \]
This simplifies to:
\[ \rho(u, v) = \frac{\sigma_x^2 - \sigma_y^2}{\sigma_x^2 + \sigma_y^2} \]

### Final Answer
Thus, the correlation coefficient between \( x+y \) and \( x-y \) is:
\[ \rho(x+y, x-y) = \frac{\sigma_x^2 - \sigma_y^2}{\sigma_x^2 + \sigma_y^2} \]