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In the given problem, we are dealing with a multivariate normal (MVN) distribution of a random vector \(\mathbf{Y} = (Y_1, Y_2, Y_3)'\) where:

\[
\mathbf{Y} \sim \text{MVN} \left( \begin{pmatrix} 4 \\ 6 \\ 10 \end{pmatrix}, \begin{pmatrix} 8 & 5 & 0 \\ 5 & 12 & 4 \\ 0 & 4 & 9 \end{pmatrix} \right)
\]

We are tasked with finding the distribution of the linear combination \( X_1 = Y_1 - Y_2 + Y_3 \).

### Step 1: Distribution of a Linear Combination of MVN Variables
If \(\mathbf{Y} \sim \text{MVN}(\mu_Y, \Sigma_Y)\), and \(X_1 = \mathbf{a}^T \mathbf{Y}\) is a linear combination of the elements of \(\mathbf{Y}\), where \(\mathbf{a} = [1, -1, 1]\), then:

\[
X_1 \sim \mathcal{N}( \mu_{X_1}, \sigma^2_{X_1})
\]

#### Mean of \(X_1\)
The mean of \(X_1\) is given by:

\[
\mu_{X_1} = \mathbf{a}^T \mu_Y = [1, -1, 1] \begin{pmatrix} 4 \\ 6 \\ 10 \end{pmatrix}
\]

\[
\mu_{X_1} = 1 \cdot 4 - 1 \cdot 6 + 1 \cdot 10 = 4 - 6 + 10 = 8
\]

#### Variance of \(X_1\)
The variance of \(X_1\) is given by:

\[
\sigma^2_{X_1} = \mathbf{a}^T \Sigma_Y \mathbf{a}
\]

Let's compute this in detail. We have:

\[
\Sigma_Y = \begin{pmatrix} 8 & 5 & 0 \\ 5 & 12 & 4 \\ 0 & 4 & 9 \end{pmatrix}
\]

And \(\mathbf{a} = [1, -1, 1]\).

First, let's compute \(\Sigma_Y \mathbf{a}\):

\[
\Sigma_Y \mathbf{a} = \begin{pmatrix} 8 & 5 & 0 \\ 5 & 12 & 4 \\ 0 & 4 & 9 \end{pmatrix} \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}
= \begin{pmatrix} 8 \cdot 1 + 5 \cdot (-1) + 0 \cdot 1 \\ 5 \cdot 1 + 12 \cdot (-1) + 4 \cdot 1 \\ 0 \cdot 1 + 4 \cdot (-1) + 9 \cdot 1 \end{pmatrix}
= \begin{pmatrix} 8 - 5 + 0 \\ 5 - 12 + 4 \\ 0 - 4 + 9 \end{pmatrix}
= \begin{pmatrix} 3 \\ -3 \\ 5 \end{pmatrix}
\]

Now, compute \(\mathbf{a}^T (\Sigma_Y \mathbf{a})\):

\[
\mathbf{a}^T \begin{pmatrix} 3 \\ -3 \\ 5 \end{pmatrix} = 1 \cdot 3 + (-1) \cdot (-3) + 1 \cdot 5 = 3 + 3 + 5 = 11
\]

Thus, the variance of \(X_1\) is \(\sigma^2_{X_1} = 11\).

### Final Distribution of \(X_1\)

\[
X_1 \sim \mathcal{N}(8, 11)
\]

Would you like further details on how the calculations were done, or do you have any additional questions?

### Related Questions:
1. How do you compute the covariance between two different linear combinations of multivariate normal variables?
2. What is the role of the covariance matrix in multivariate normal distributions?
3. How does the structure of the covariance matrix affect the independence of the components in \(\mathbf{Y}\)?
4. What changes if we introduce additional linear combinations to the problem?
5. How does the distribution of \(X_1\) compare to other linear combinations of \(\mathbf{Y}\)?

### Tip:
When dealing with linear combinations of multivariate normal distributions, always check if the covariance matrix has any special structure, such as block-diagonal, which can simplify computations.

Find the distribution of X1 = Y1 - Y2 + Y3, where Y = (Y1, Y2, Y3)' follows a multivariate normal distribution with mean vector [4, 6, 10]' and covariance matrix [[8, 5, 0], [5, 12, 4], [0, 4, 9]].

Learn how to find the distribution of X1 = Y1 - Y2 + Y3 when Y follows a multivariate normal distribution with a specified mean vector and covariance matrix. This involves calculating the mean and variance of the linear combination of the variables.

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