Two variates X and Y have zero means, the same variance `sigma^(2)` and zero correlation. Then `U=Xcosalpha+Ysinalpha,V=Xsinalpha-Ycosalpha` have correlation

Options

A. `-1`
B. 1
C. 0
D. `(1)/(2)`

Answer

To solve the problem, we need to find the correlation between the two new variates \( U \) and \( V \) defined as:

\[ U = X \cos \alpha + Y \sin \alpha \]
\[ V = X \sin \alpha - Y \cos \alpha \]

Given that \( X \) and \( Y \) have zero means, the same variance \( \sigma^2 \), and zero correlation, we can use the properties of covariance to find the correlation between \( U \) and \( V \).

### Step-by-Step Solution:

1. **Calculate the Covariance of \( U \) and \( V \)**:
The covariance of two linear combinations of random variables can be expressed using the formula:
\[ \text{Cov}(aX + bY, cX + dY) = ac \text{Var}(X) + ad \text{Cov}(X, Y) + bc \text{Cov}(Y, X) + bd \text{Var}(Y) \]
Here, we have:
- \( a = \cos \alpha \), \( b = \sin \alpha \)
- \( c = \sin \alpha \), \( d = -\cos \alpha \)

Therefore, we can write:
\[ \text{Cov}(U, V) = \cos \alpha \cdot \sin \alpha \cdot \text{Var}(X) + \cos \alpha \cdot (-\cos \alpha) \cdot \text{Cov}(X, Y) + \sin \alpha \cdot \sin \alpha \cdot \text{Cov}(Y, X) + \sin \alpha \cdot (-\cos \alpha) \cdot \text{Var}(Y) \]

2. **Substituting Known Values**:
Since \( \text{Var}(X) = \text{Var}(Y) = \sigma^2 \) and \( \text{Cov}(X, Y) = 0 \) (because they are uncorrelated), we can simplify:
\[ \text{Cov}(U, V) = \cos \alpha \cdot \sin \alpha \cdot \sigma^2 + 0 + 0 - \sin \alpha \cdot \cos \alpha \cdot \sigma^2 \]
This simplifies to:
\[ \text{Cov}(U, V) = \sigma^2 (\cos \alpha \sin \alpha - \sin \alpha \cos \alpha) = 0 \]

3. **Calculate the Variances of \( U \) and \( V \)**:
Now we need to find the variances of \( U \) and \( V \):
- For \( U \):
\[ \text{Var}(U) = \text{Var}(X \cos \alpha + Y \sin \alpha) = \cos^2 \alpha \text{Var}(X) + \sin^2 \alpha \text{Var}(Y) + 2 \cos \alpha \sin \alpha \text{Cov}(X, Y) \]
Substituting the known values:
\[ \text{Var}(U) = \cos^2 \alpha \sigma^2 + \sin^2 \alpha \sigma^2 + 0 = \sigma^2 (\cos^2 \alpha + \sin^2 \alpha) = \sigma^2 \]

- For \( V \):
\[ \text{Var}(V) = \text{Var}(X \sin \alpha - Y \cos \alpha) = \sin^2 \alpha \text{Var}(X) + \cos^2 \alpha \text{Var}(Y) - 2 \sin \alpha \cos \alpha \text{Cov}(X, Y) \]
Substituting the known values:
\[ \text{Var}(V) = \sin^2 \alpha \sigma^2 + \cos^2 \alpha \sigma^2 + 0 = \sigma^2 (\sin^2 \alpha + \cos^2 \alpha) = \sigma^2 \]

4. **Calculate the Correlation**:
The correlation \( r \) between \( U \) and \( V \) is given by:
\[ r = \frac{\text{Cov}(U, V)}{\sqrt{\text{Var}(U) \cdot \text{Var}(V)}} \]
Substituting the values we found:
\[ r = \frac{0}{\sqrt{\sigma^2 \cdot \sigma^2}} = \frac{0}{\sigma^2} = 0 \]

Thus, the correlation between \( U \) and \( V \) is **0**.

### Final Answer:
The correlation between \( U \) and \( V \) is **0**.