The question in the image reads:

"From a data set with two variables $x_1$ and $x_2$, the covariance matrix is given by: $\begin{bmatrix} 1.6 & 0.6 \\ 0.6 & 0.9 \end{bmatrix}$ Find $\rho$."

To solve this, we need to determine $\rho$, which represents the correlation coefficient between the variables $x_1$ and $x_2$. Here’s the step-by-step process:

Step 1: Recall the Covariance Matrix Structure

A covariance matrix for two variables $x_1$ and $x_2$ is generally given by: $\begin{bmatrix} \sigma_{x_1}^2 & \text{Cov}(x_1, x_2) \\ \text{Cov}(x_1, x_2) & \sigma_{x_2}^2 \end{bmatrix}$ where:

• $\sigma_{x_1}^2$ is the variance of $x_1$,
• $\sigma_{x_2}^2$ is the variance of $x_2$,
• $\text{Cov}(x_1, x_2)$ is the covariance between $x_1$ and $x_2$.

Step 2: Identify Values from the Matrix

From the matrix: $\begin{bmatrix} 1.6 & 0.6 \\ 0.6 & 0.9 \end{bmatrix}$ we can identify:

• $\sigma_{x_1}^2 = 1.6$ (variance of $x_1$),
• $\sigma_{x_2}^2 = 0.9$ (variance of $x_2$),
• $\text{Cov}(x_1, x_2) = 0.6$.

Step 3: Calculate Standard Deviations

The standard deviation of $x_1$ (denoted $\sigma_{x_1}$) and $x_2$ (denoted $\sigma_{x_2}$) can be found by taking the square roots of the variances: $\sigma_{x_1} = \sqrt{1.6} \approx 1.2649$ $\sigma_{x_2} = \sqrt{0.9} \approx 0.9487$

Step 4: Use the Formula for Correlation Coefficient $\rho$

The correlation coefficient $\rho$ between $x_1$ and $x_2$ is given by: $\rho = \frac{\text{Cov}(x_1, x_2)}{\sigma_{x_1} \sigma_{x_2}}$ Substitute the values: $\rho = \frac{0.6}{1.2649 \times 0.9487}$

Step 5: Perform the Calculation

Calculate the denominator: $1.2649 \times 0.9487 \approx 1.2$ Then divide: $\rho = \frac{0.6}{1.2} = 0.5$

Final Answer

The correlation coefficient $\rho$ between $x_1$ and $x_2$ is: $\rho = 0.5$

Would you like further details or have questions about any part of this solution?

Here are some related questions you might find useful:

1. How is a covariance matrix constructed for multiple variables?
2. What does a correlation coefficient signify in statistical analysis?
3. How does the covariance matrix relate to data spread and variance?
4. Can a covariance matrix have negative values? What does that imply?
5. How would we interpret $\rho$ if it were close to 1 or -1?

Tip: Covariance indicates the direction of the relationship between two variables, while correlation also measures the strength of the relationship.