We are given a bivariate Gaussian distribution with:

• Mean vector $\mu^T = [1 \ 2]$
• Covariance matrix $\Sigma = \begin{bmatrix} 4 & -2 \\ -2 & 9 \end{bmatrix}$

We need to find $a + b$, where:

• $a = \mathbb{E}[X_1]$ (the expected value of $X_1$),
• $b = \sigma_{X_1}$ (the standard deviation of $X_1$).

Step 1: Find $a = \mathbb{E}[X_1]$

The expected value of $X_1$ is the first element of the mean vector $\mu$. Given $\mu^T = [1 \ 2]$, we have: $a = \mathbb{E}[X_1] = 1$

Step 2: Find $b = \sigma_{X_1}$

The standard deviation $\sigma_{X_1}$ is the square root of the variance of $X_1$. The variance of $X_1$ is the (1,1) element of the covariance matrix $\Sigma$. From $\Sigma = \begin{bmatrix} 4 & -2 \\ -2 & 9 \end{bmatrix}$, the variance of $X_1$ is 4.

Thus, the standard deviation is: $b = \sigma_{X_1} = \sqrt{4} = 2$

Step 3: Find $a + b$

Now, we can calculate $a + b$: $a + b = 1 + 2 = 3$

Thus, $a + b = 3$.

Would you like more details on any part of the solution? Here are some related questions for deeper understanding:

1. How is the covariance matrix used in multivariate Gaussian distributions?
2. How can we calculate the covariance between $X_1$ and $X_2$ from $\Sigma$?
3. What is the interpretation of the mean vector $\mu$ in a multivariate Gaussian distribution?
4. What are the properties of the multivariate normal distribution?
5. How do the eigenvalues of $\Sigma$ relate to the distribution's properties?

Tip: In a multivariate Gaussian distribution, the diagonal elements of the covariance matrix represent the variances of individual variables, while the off-diagonal elements represent the covariances between variables.