The two variates X and Y are uncorrelated and have standard deviations `sigma_(X) and sigma_(Y)` respectively, the correlation coefficient between `X+Y` and `X-Y` is

Options

A. `(sigma_(X)sigma_(Y))/(sigma_(X)^(2)+sigma_(Y)^(2))`
B. `(1)/(2)((1)/(sigma_(X))+(1)/(sigma_(Y)))`
C. `(sigma_(X)^(2)-sigma_(Y)^(2))/(sigma_(X)^(2)+sigma_(Y)^(2))`
D. none of these

Answer

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

### Step 1: Understand the properties of covariance and variance
We know that the correlation coefficient \(r\) between two variables \(A\) and \(B\) can be expressed as:
\[ r(A, B) = \frac{\text{Cov}(A, B)}{\sqrt{\text{Var}(A) \cdot \text{Var}(B)}} \]
where \(\text{Cov}(A, B)\) is the covariance between \(A\) and \(B\), and \(\text{Var}(A)\) and \(\text{Var}(B)\) are the variances of \(A\) and \(B\) respectively.

### Step 2: Define \(A\) and \(B\)
In our case, let:
- \(A = X + Y\)
- \(B = X - Y\)

### Step 3: Calculate the covariance \(\text{Cov}(A, B)\)
Using the property of covariance:
\[ \text{Cov}(A, B) = \text{Cov}(X + Y, X - Y) \]
This can be expanded using the linearity of covariance:
\[ \text{Cov}(A, B) = \text{Cov}(X, X) - \text{Cov}(X, Y) + \text{Cov}(Y, X) - \text{Cov}(Y, Y) \]
Since \(X\) and \(Y\) are uncorrelated, \(\text{Cov}(X, Y) = 0\) and \(\text{Cov}(Y, X) = 0\). Therefore:
\[ \text{Cov}(A, B) = \text{Var}(X) - \text{Var}(Y) = \sigma_X^2 - \sigma_Y^2 \]

### Step 4: Calculate the variances \(\text{Var}(A)\) and \(\text{Var}(B)\)
Now we need to find the variances of \(A\) and \(B\):
1. For \(A = X + Y\):
\[ \text{Var}(A) = \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) = \sigma_X^2 + \sigma_Y^2 \]
2. For \(B = X - Y\):
\[ \text{Var}(B) = \text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) = \sigma_X^2 + \sigma_Y^2 \]

### Step 5: Substitute into the correlation formula
Now we can substitute the values into the correlation coefficient formula:
\[ r(A, B) = \frac{\text{Cov}(A, B)}{\sqrt{\text{Var}(A) \cdot \text{Var}(B)}} \]
Substituting the calculated values:
\[ r(A, B) = \frac{\sigma_X^2 - \sigma_Y^2}{\sqrt{(\sigma_X^2 + \sigma_Y^2)(\sigma_X^2 + \sigma_Y^2)}} \]
This simplifies to:
\[ r(A, B) = \frac{\sigma_X^2 - \sigma_Y^2}{\sigma_X^2 + \sigma_Y^2} \]

### Final Answer
Thus, the correlation coefficient between \(X + Y\) and \(X - Y\) is:
\[ \frac{\sigma_X^2 - \sigma_Y^2}{\sigma_X^2 + \sigma_Y^2} \]