X and Y are two correlated variables with the same standard deviation and the correlation coefficient r, the correlation coefficient between X and `X+Y` is

Options

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

Answer

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

### Step 1: Understand the variables
We are given two correlated variables \(X\) and \(Y\) with the same standard deviation. Let's denote the standard deviation of both \(X\) and \(Y\) as \(\sigma\). The correlation coefficient between \(X\) and \(Y\) is given as \(r\).

### Step 2: Define the correlation coefficient
The correlation coefficient \(R_{X, X+Y}\) can be defined as:
\[ R_{X, X+Y} = \frac{\text{Cov}(X, X+Y)}{\sigma_X \cdot \sigma_{X+Y}} \]
where \(\text{Cov}(X, X+Y)\) is the covariance between \(X\) and \(X + Y\), and \(\sigma_X\) and \(\sigma_{X+Y}\) are the standard deviations of \(X\) and \(X + Y\) respectively.

### Step 3: Calculate the covariance
To find \(\text{Cov}(X, X + Y)\):
\[ \text{Cov}(X, X + Y) = \text{Cov}(X, X) + \text{Cov}(X, Y) = \text{Var}(X) + \text{Cov}(X, Y) \]
Since \(\text{Var}(X) = \sigma^2\) and \(\text{Cov}(X, Y) = r \cdot \sigma^2\), we can substitute:
\[ \text{Cov}(X, X + Y) = \sigma^2 + r \cdot \sigma^2 = \sigma^2(1 + r) \]

### Step 4: Calculate the variance of \(X + Y\)
Next, we need to find the variance of \(X + Y\):
\[ \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y) = \sigma^2 + \sigma^2 + 2r \cdot \sigma^2 = 2\sigma^2(1 + r) \]
Thus, the standard deviation of \(X + Y\) is:
\[ \sigma_{X + Y} = \sqrt{\text{Var}(X + Y)} = \sqrt{2\sigma^2(1 + r)} = \sigma \sqrt{2(1 + r)} \]

### Step 5: Substitute into the correlation coefficient formula
Now we can substitute back into the correlation coefficient formula:
\[ R_{X, X+Y} = \frac{\text{Cov}(X, X+Y)}{\sigma_X \cdot \sigma_{X+Y}} = \frac{\sigma^2(1 + r)}{\sigma \cdot \sigma \sqrt{2(1 + r)}} \]
This simplifies to:
\[ R_{X, X+Y} = \frac{(1 + r)}{\sqrt{2(1 + r)}} = \frac{\sqrt{1 + r}}{\sqrt{2}} = \frac{\sqrt{1 + r}}{\sqrt{2}} \]

### Final Result
Thus, the correlation coefficient between \(X\) and \(X + Y\) is:
\[ R_{X, X+Y} = \frac{\sqrt{1 + r}}{\sqrt{2}} \]