The correlation between X and `a-X` is

Options

A. `-1`
B. `-(1)/(2)`
C. `-(1)/(4)`
D. 0

Answer

To find the correlation between \( X \) and \( a - X \), we can follow these steps:

### Step 1: Understand the correlation formula
The correlation coefficient \( r \) between two variables \( X \) and \( Y \) is given by the formula:
\[ r_{XY} = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X)} \cdot \sqrt{\text{Var}(Y)}} \]
In our case, we want to find \( r_{X, a-X} \).

### Step 2: Identify the variables
Here, we have:
- \( Y = a - X \)

### Step 3: Calculate the covariance
The covariance between \( X \) and \( a - X \) can be calculated as follows:
\[ \text{Cov}(X, a - X) = \text{Cov}(X, a) - \text{Cov}(X, X) \]
Since \( a \) is a constant, \( \text{Cov}(X, a) = 0 \). Therefore:
\[ \text{Cov}(X, a - X) = 0 - \text{Var}(X) = -\text{Var}(X) \]

### Step 4: Calculate the variance of \( a - X \)
Next, we calculate the variance of \( a - X \):
\[ \text{Var}(a - X) = \text{Var}(X) \quad \text{(since variance is unaffected by a constant)} \]

### Step 5: Substitute into the correlation formula
Now we can substitute the covariance and variances into the correlation formula:
\[ r_{X, a-X} = \frac{-\text{Var}(X)}{\sqrt{\text{Var}(X)} \cdot \sqrt{\text{Var}(X)}} \]
This simplifies to:
\[ r_{X, a-X} = \frac{-\text{Var}(X)}{\text{Var}(X)} = -1 \]

### Step 6: Conclusion
Thus, the correlation between \( X \) and \( a - X \) is:
\[ \boxed{-1} \]