State True or False: corr(`x,x`) = `0`

Answer

To determine whether the statement "corr(x, x) = 0" is true or false, we need to analyze the correlation of a variable with itself.

### Step-by-Step Solution:

1. **Understanding Correlation**:
The correlation coefficient, denoted as corr(X, Y), measures the strength and direction of a linear relationship between two variables X and Y. The formula for the correlation coefficient is:
\[ \text{corr}(X, Y) = \frac{\text{cov}(X, Y)}{\sqrt{\text{var}(X) \cdot \text{var}(Y)}} \]

2. **Covariance of X with Itself**:
When we calculate the covariance of a variable with itself, we have:
\[ \text{cov}(X, X) = E(X \cdot X) - E(X) \cdot E(X) = E(X^2) - (E(X))^2 \]
This expression is actually the definition of the variance of X:
\[ \text{var}(X) = E(X^2) - (E(X))^2 \]

3. **Substituting into the Correlation Formula**:
Now, substituting into the correlation formula:
\[ \text{corr}(X, X) = \frac{\text{cov}(X, X)}{\sqrt{\text{var}(X) \cdot \text{var}(X)}} \]
Since \(\text{cov}(X, X) = \text{var}(X)\), we can rewrite this as:
\[ \text{corr}(X, X) = \frac{\text{var}(X)}{\sqrt{\text{var}(X) \cdot \text{var}(X)}} \]

4. **Simplifying the Expression**:
The denominator simplifies to:
\[ \sqrt{\text{var}(X) \cdot \text{var}(X)} = \text{var}(X) \]
Therefore, we have:
\[ \text{corr}(X, X) = \frac{\text{var}(X)}{\text{var}(X)} = 1 \]

5. **Conclusion**:
Since we have established that \(\text{corr}(X, X) = 1\), the statement "corr(x, x) = 0" is **False**.

### Final Answer:
The statement "corr(x, x) = 0" is **False**.