Let X and Y be two variables with correlation coefficient e. If the values of X and Y serires and changed such that the cov (X,Y) remains unchanged while the variance of X and Y becomes 4 times their original values, then find the new correlation coefficient between X and Y.

Answer

To find the new correlation coefficient between X and Y after the changes in their variances, we can follow these steps:

### Step 1: Understand the correlation coefficient formula
The correlation coefficient \( r \) between two variables X and Y is given by the formula:
\[ r_{X,Y} = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]
Where:
- \(\text{Cov}(X,Y)\) is the covariance between X and Y.
- \(\text{Var}(X)\) is the variance of X.
- \(\text{Var}(Y)\) is the variance of Y.

### Step 2: Define the initial conditions
Let:
- The initial correlation coefficient be \( e \).
- The initial covariance be \( \text{Cov}(X,Y) \).
- The initial variances be \( \text{Var}(X) \) and \( \text{Var}(Y) \).

From the formula, we can express the initial correlation as:
\[ e = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]

### Step 3: Analyze the changes
According to the problem:
- The covariance \( \text{Cov}(X,Y) \) remains unchanged.
- The variances of X and Y become 4 times their original values:
- New variance of X: \( \text{Var}(X)_{\text{new}} = 4 \cdot \text{Var}(X) \)
- New variance of Y: \( \text{Var}(Y)_{\text{new}} = 4 \cdot \text{Var}(Y) \)

### Step 4: Substitute into the correlation formula
Now, we can find the new correlation coefficient \( r_{\text{new}} \):
\[ r_{\text{new}} = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)_{\text{new}} \cdot \text{Var}(Y)_{\text{new}}}} \]
Substituting the new variances:
\[ r_{\text{new}} = \frac{\text{Cov}(X,Y)}{\sqrt{(4 \cdot \text{Var}(X)) \cdot (4 \cdot \text{Var}(Y))}} \]
\[ r_{\text{new}} = \frac{\text{Cov}(X,Y)}{\sqrt{16 \cdot \text{Var}(X) \cdot \text{Var}(Y)}} \]
\[ r_{\text{new}} = \frac{\text{Cov}(X,Y)}{4 \cdot \sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]

### Step 5: Relate the new correlation to the old correlation
From our initial correlation \( e \):
\[ r_{\text{new}} = \frac{1}{4} \cdot \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]
Thus,
\[ r_{\text{new}} = \frac{e}{4} \]

### Final Answer
The new correlation coefficient between X and Y is:
\[ \boxed{\frac{e}{4}} \]