The image contains a mathematical formula and a question about the correlation coefficient. Here's the content of the image and an explanation:

The formula given defines the correlation coefficient $r_{ij}$ between two random variables $X_i$ and $X_j$ as:

$r_{ij} = \frac{\sigma_{ij}}{\sigma_i \sigma_j}$

Where:

• $\sigma_{ij}$ is the covariance between $X_i$ and $X_j$,
• $\sigma_i$ is the standard deviation of $X_i$,
• $\sigma_j$ is the standard deviation of $X_j$.

The question then asks: What is the value of $r_{ii}$?

Explanation:

The correlation coefficient $r_{ii}$ corresponds to the case where $i = j$. In this case:

• $\sigma_{ii}$ is the covariance of $X_i$ with itself, which is the variance of $X_i$, denoted as $\sigma_i^2$.
• $\sigma_i \sigma_j$ becomes $\sigma_i \cdot \sigma_i = \sigma_i^2$.

Thus:

$r_{ii} = \frac{\sigma_{ii}}{\sigma_i^2} = \frac{\sigma_i^2}{\sigma_i^2} = 1$

Conclusion:

The value of $r_{ii}$ is 1. This is because the correlation of any random variable with itself is always 1.

Let me know if you'd like more details on this or have any questions!

Here are 5 related questions:

1. What does a correlation coefficient of 0 indicate?
2. How is covariance related to correlation?
3. What happens to $r_{ij}$ if $X_i$ and $X_j$ are independent?
4. Can the correlation coefficient ever be greater than 1?
5. How do we interpret a negative correlation coefficient?

Tip: The correlation matrix is always symmetric, and all diagonal elements are 1 since each variable is perfectly correlated with itself.