Let X be a 3-dimensional random vector with dispersion matrix  

Determine the first principal component and the proportion of the total variability that it explains.

To determine the first principal component of the random vector \( X \) and the proportion of total variability it explains, we need to perform principal component analysis (PCA) on the dispersion matrix \( \Sigma \).

The first principal component, denoted by \( v_1 \), corresponds to the eigenvector associated with the largest eigenvalue of \( \Sigma \). The proportion of total variability explained by the first principal component is given by the ratio of the largest eigenvalue to the sum of all eigenvalues.

Let's calculate it step by step:

1. First, we need to find the eigenvalues and eigenvectors of the dispersion matrix \( \Sigma \).
2. Then, we will identify the eigenvector corresponding to the largest eigenvalue, which will be our first principal component.
3. Finally, we will calculate the proportion of total variability explained by the first principal component.

Given \( \Sigma \):
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