Let X and Y be two variables with the same variance and let `U=X+Y,V=X-Y` then cov (U, V) is equal to

Options

A. cov (X, Y)
B. 0
C. 1
D. `-1`

Answer

To find the covariance \( \text{cov}(U, V) \) where \( U = X + Y \) and \( V = X - Y \), we can use the formula for covariance:

\[ \text{cov}(aX + bY, cX + dY) = ac \cdot \text{var}(X) + ad \cdot \text{cov}(X, Y) + bc \cdot \text{cov}(Y, X) + bd \cdot \text{var}(Y) \]

### Step 1: Identify the coefficients
In our case:
- \( U = X + Y \) corresponds to \( a = 1 \), \( b = 1 \)
- \( V = X - Y \) corresponds to \( c = 1 \), \( d = -1 \)

### Step 2: Substitute into the covariance formula
Now, substituting these values into the covariance formula:

\[ \text{cov}(U, V) = (1)(1) \cdot \text{var}(X) + (1)(-1) \cdot \text{cov}(X, Y) + (1)(1) \cdot \text{cov}(Y, X) + (1)(-1) \cdot \text{var}(Y) \]

### Step 3: Simplify the expression
Since \( \text{cov}(Y, X) = \text{cov}(X, Y) \), we can denote \( \text{cov}(X, Y) \) as \( \text{cov}(X, Y) = C \). Also, we know that \( \text{var}(X) = \text{var}(Y) = \sigma^2 \) (given that both have the same variance).

Substituting these into the equation gives:

\[ \text{cov}(U, V) = 1 \cdot \sigma^2 + (-1) \cdot C + 1 \cdot C + (-1) \cdot \sigma^2 \]

### Step 4: Combine like terms
Now, combine the terms:

\[ \text{cov}(U, V) = \sigma^2 - C + C - \sigma^2 \]

The \( \sigma^2 \) terms cancel out, and the \( C \) terms also cancel out:

\[ \text{cov}(U, V) = 0 \]

### Conclusion
Thus, the covariance \( \text{cov}(U, V) \) is equal to \( 0 \).

### Final Answer
The answer is \( 0 \).

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