Let's break down the problem and work through the steps needed to find the solutions.

Problem Summary

We are given:

• Two Gaussian random variables $X_1$ and $X_2$ with mean vector: $\mathbf{X} = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$ and covariance matrix: $C_{\mathbf{X}} = \begin{bmatrix} 5 & -\frac{2}{\sqrt{5}} \\ -\frac{2}{\sqrt{5}} & 4 \end{bmatrix}$

• A transformation matrix $T$ to form new random variables $Y_1$ and $Y_2$, where: $T = \begin{bmatrix} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{bmatrix}$

We are tasked to:

1. Find the mean vector $\mathbf{Y}$ for the transformed variables.
2. Find the covariance matrix $C_{\mathbf{Y}}$.
3. Find the correlation coefficient between $Y_1$ and $Y_2$.

Step 1: Mean vector of $\mathbf{Y}$

The mean of the new random variables $\mathbf{Y}$ is found using the transformation matrix: $\mathbf{Y} = T \mathbf{X}$ where $\mathbf{X}$ is the mean vector of the original random variables.

Step 2: Covariance matrix of $\mathbf{Y}$

The covariance matrix $C_{\mathbf{Y}}$ is found using the formula: $C_{\mathbf{Y}} = T C_{\mathbf{X}} T^T$ where $C_{\mathbf{X}}$ is the covariance matrix of the original variables, and $T^T$ is the transpose of the transformation matrix $T$.

Step 3: Correlation coefficient between $Y_1$ and $Y_2$

The correlation coefficient $\rho(Y_1, Y_2)$ is given by: $\rho(Y_1, Y_2) = \frac{C_{\mathbf{Y}}(1,2)}{\sqrt{C_{\mathbf{Y}}(1,1) C_{\mathbf{Y}}(2,2)}}$ where $C_{\mathbf{Y}}(1,2)$ is the covariance between $Y_1$ and $Y_2$, and $C_{\mathbf{Y}}(1,1)$ and $C_{\mathbf{Y}}(2,2)$ are the variances of $Y_1$ and $Y_2$, respectively.

I will now perform these calculations.### Results:

1. Mean vector of $\mathbf{Y}$: $\mathbf{Y} = \begin{bmatrix} 1.5 \\ 0 \end{bmatrix}$

2. Covariance matrix of $C_{\mathbf{Y}}$: $C_{\mathbf{Y}} = \begin{bmatrix} 5.106 & 3.382 \\ 3.382 & 4.356 \end{bmatrix}$

3. Correlation coefficient between $Y_1$ and $Y_2$: $\rho(Y_1, Y_2) \approx 0.717$

Would you like more details or explanations on any of these steps?

Here are some related questions for further exploration:

1. How do you interpret the covariance matrix in terms of variance and covariance between variables?
2. What would happen if you used a different transformation matrix $T$?
3. How is the correlation coefficient related to covariance?
4. What properties of Gaussian random variables simplify these calculations?
5. How could you generalize this process for more than two random variables?

Tip: When computing covariance transformations, ensure that your transformation matrix $T$ is invertible to maintain consistent results.