If X and Y are two independent variables with `sigma_(X)^(2)=36,sigma_(Y)^(2)=16` then correlation coefficient between `U=X+Y and V=X-Y` is

Options

A. `(5)/(13)`
B. `-(5)/(13)`
C. `(4)/(9)`
D. none of these

Answer

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

### Step 1: Understand the given information
We have two independent random variables \( X \) and \( Y \) with variances:
- \( \sigma_X^2 = 36 \)
- \( \sigma_Y^2 = 16 \)

### Step 2: Calculate the covariance of \( U \) and \( V \)
The correlation coefficient \( r \) is given by the formula:
\[ r = \frac{\text{Cov}(U, V)}{\sqrt{\text{Var}(U) \cdot \text{Var}(V)}} \]
where \( U = X + Y \) and \( V = X - Y \).

#### Covariance Calculation
Using the properties of covariance:
\[ \text{Cov}(U, V) = \text{Cov}(X + Y, X - Y) \]
Using the linearity of covariance:
\[ \text{Cov}(U, V) = \text{Cov}(X, X) - \text{Cov}(X, Y) + \text{Cov}(Y, X) - \text{Cov}(Y, Y) \]
Since \( X \) and \( Y \) are independent, \( \text{Cov}(X, Y) = 0 \) and \( \text{Cov}(Y, X) = 0 \). Thus:
\[ \text{Cov}(U, V) = \text{Var}(X) - \text{Var}(Y) = 36 - 16 = 20 \]

### Step 3: Calculate the variances of \( U \) and \( V \)
#### Variance of \( U \)
\[ \text{Var}(U) = \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) = 36 + 16 = 52 \]

#### Variance of \( V \)
\[ \text{Var}(V) = \text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) = 36 + 16 = 52 \]

### Step 4: Substitute into the correlation coefficient formula
Now we have:
\[ r = \frac{\text{Cov}(U, V)}{\sqrt{\text{Var}(U) \cdot \text{Var}(V)}} = \frac{20}{\sqrt{52 \cdot 52}} = \frac{20}{52} = \frac{5}{13} \]

### Final Answer
The correlation coefficient between \( U \) and \( V \) is:
\[ \boxed{\frac{5}{13}} \]