91Ó°ÊÓ

A linear combination is an expression made up of sums and scalar multiplications. In the context of multivariate normal distributions, it often involves combining multiple random variables to create a new variable. When we talk about \(\bar{X} = n^{-1} \sum_{i=1}^{n} X_{i}\), we're essentially creating a linear combination of the components of \(\mathbf{X}\). Here, \(\bar{X}\) represents the arithmetic mean of the components, expressed as \(\mathbf{a^\intercal X}\).

• \(\mathbf{a}\) is a vector where each element is \(1/n\).
• You multiply each component of \(\mathbf{X}\) by \(1/n\) and sum these up.

This formulation not only simplifies calculations but is crucial when finding the distribution of \(\bar{X}\). By re-expressing \(\bar{X}\) as a linear combination, we can apply various mathematical rules (like Theorem 3.5.1) to deduce properties such as its mean and variance.

The covariance matrix \(\mathbf{\Sigma}\) captures how each pair of elements in the random vector \(\mathbf{X}\) interact with each other. Each entry \(\sigma_{ij}\) in the matrix gives the covariance between the \(i\text{th}\) and \(j\text{th}\) elements of \(\mathbf{X}\). This concept is vital when analyzing the spread or variability of the data represented by \(\mathbf{X}\).

• If \(\mathbf{X}\) has independent components, off-diagonal entries (\(\sigma_{ij}, i eq j\)) will be zero.
• The diagonal entries represent the variance of each component.

When we apply a linear transformation on \(\mathbf{X}\), like forming \(\bar{X}\) as \(\mathbf{a^\intercal X}\), the variance of \(\bar{X}\) can be calculated using the formula \(\mathbf{a^\intercal \Sigma a}\).This is essential for finding the variance of \(\bar{X}\) once it's expressed as a linear combination.

A mean vector \(\boldsymbol{\mu}\) in a multivariate distribution contains the expected values (means) of each of the random variables included in \(\mathbf{X}\). For a random vector \(\mathbf{X}\) with \(n\) components, the mean vector is \((\mu_{1}, \mu_{2}, \ldots, \mu_{n})\).1. **Importance**: - This vector indicates the central tendency of the entire multivariate distribution.2. **Affect on \(\bar{X}\)**: - When calculating \(\bar{X}\) as \(\mathbf{a^\intercal X}\), the mean of \(\bar{X}\) becomes \(\mathbf{a^\intercal \boldsymbol{\mu}}\).This simplifies to \(\mu/n\) if all elements of \(\boldsymbol{\mu}\) are the same, signifying that \(\bar{X}\) has the same mean as any individual component of \(\mathbf{X}\) under certain conditions. Knowing the mean vector is crucial in identifying the intrinsic characteristics of \(\bar{X}\).