2.4 Random Vectors and Matrices

For a set of random variables, we can model the joint distribution of those random variables.

Matrix notation can help us organize distributional information about the set of random variables.

2.4.1 Random Vectors

A random vector is a vector (a collection) of \(m\) indexed random variables such that \[\mathbf{X} = \left(\begin{array}{c} X_{1}\\ X_{2}\\ \vdots\\\\ X_{m}\\ \end{array}\right)\]

The expected value of a random vector \(\mathbf{X}\), written as \(E(\mathbf{X})\), is a vector of expected values, \[\boldsymbol{\mu} = E(\mathbf{X}) = \left(\begin{array}{c} E(X_1)\\ E(X_2)\\ \vdots\\ E(X_m) \end{array} \right)= \left(\begin{array}{c} \mu_1\\ \mu_2\\ \vdots\\ \mu_m \end{array} \right) \] where \(E(X_1) = \mu_1\),…,\(E(X_m) = \mu_m\)

The covariances of all pairs of values in a random vector \(\mathbf{X}\), can be organized in a covariance matrix,

\[\boldsymbol{\Sigma} = Cov(\mathbf{X}) = \left(\begin{array}{cccc}\sigma^2_1&\sigma_{12}&\cdots&\sigma_{1m}\\\sigma_{21}&\sigma^2_2&\cdots&\sigma_{2m}\\\vdots&\vdots&\ddots&\vdots\\\sigma_{m1}&\sigma_{m2}&\cdots&\sigma^2_m\\ \end{array} \right) \]

using the notation introduced above that \(\sigma^2_l = Var(X_l)\) and \(\sigma_{lk} = Cov(X_l,X_k)\). We’ll talk much more about the covariance matrix.

2.4.2 Random Matrices

Let \(X_{ij}\) be an indexed random variable where index \(i=1,...,n\) refers to different subjects (different units) and the index \(j = 1,...,m\) refers to different observations over time or space. We can organize these variables in random vectors and then into a matrix for convenience.

The random matrix \(\mathbf{X}\) is written as \[\mathbf{X} = \left(\begin{array}{cccc} X_{11}&X_{12}&\cdots&X_{1m}\\ X_{21}&X_{22}&\cdots&X_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ X_{n1}&X_{n2}&\cdots&X_{nm}\\ \end{array}\right) \] in which each column represents one observation time and the rows represent the \(n\) subjects or individual units.

The expected value of a random matrix \(\mathbf{X}\), written as \(E(\mathbf{X})\), is a matrix of expected values, \[E(\mathbf{X}) = \left(\begin{array}{cccc} E(X_{11})&E(X_{12})&\cdots&E(X_{1m})\\ E(X_{21})&E(X_{22})&\cdots&E(X_{2m})\\ \vdots&\vdots&\ddots&\vdots\\ E(X_{n1})&E(X_{n2})&\cdots&E(X_{nm}) \end{array} \right)\]

2.4.2.1 Properties

For random matrices or vectors of the same size \(\mathbf{X}\) and \(\mathbf{Y}\) and constant (not random) compatible matrices or vectors \(\mathbf{A}\) and \(\mathbf{B}\), the following are true,

\[E(\mathbf{X}+\mathbf{Y})=E(\mathbf{X})+E(\mathbf{Y})\] \[E(\mathbf{A}\mathbf{X}\mathbf{B})=\mathbf{A}E(\mathbf{X})\mathbf{B}\]

• Note: We will use bold typeface for random vectors and matrices and normal typeface for random variables.