class: center, middle, inverse, title-slide # STA 610L: Review ## Random vectors and matrices ### Dr. Olanrewaju Michael Akande --- ## Random vector Suppose `\begin{eqnarray*} {\bf Y}=\begin{pmatrix} Y_1 \\ Y_2 \\ \vdots \\ Y_n \end{pmatrix} \end{eqnarray*}` is a vector of random variables with `\(E(Y_i)=\mu_i\)`, `\(\text{Var}(Y_i)=\sigma_{ii}\)`, and `\(\text{Cov}(Y_i,Y_j)=\sigma_{ij}\)`. --- ## Expectation of a vector The expectation of the random vector `\({\bf Y}\)` is defined `\begin{eqnarray*} E({\bf Y})=\begin{pmatrix} E(Y_1) \\ E(Y_2) \\ \vdots \\ E(Y_n) \end{pmatrix} = \begin{pmatrix} \mu_1 \\ \mu_2 \\ \vdots \\ \mu_n \end{pmatrix} = \mu. \end{eqnarray*}` --- ## Expectation of a matrix Suppose `\({\bf Z}\)` is an `\((n \times p)\)` matrix of random variables. Then `\begin{eqnarray*} E({\bf Z})=\begin{pmatrix} E(Z_{11}) & \cdots & E(Z_{1p}) \\ \vdots & \cdots & \vdots \\ E(Z_{n1}) & \cdots & E(Z_{np}) \end{pmatrix}. \end{eqnarray*}` Thus the expectation of a random matrix is the matrix of the expectations. --- ## Covariance For `\({\bf Y}\)` an `\((n \times 1)\)` random vector, the *covariance matrix* of `\({\bf Y}\)` is defined as `\begin{eqnarray*} \text{Cov}({\bf Y} )&=&E\left[({\bf Y}-\mu)({\bf Y}-\mu)' \right] \\ &=& \Sigma = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \cdots & \sigma_{1n} \\ \sigma_{21} & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots \\ \sigma_{n1} & \cdots & \cdots & \sigma_{nn} \end{pmatrix}, \end{eqnarray*}` where `\(\sigma_{ij}=E[(Y_i -\mu_i)(Y_j-\mu_j)]\)`, `\(i,j=1,\ldots,n\)`. --- ## Linear combinations Suppose `\({\bf Y}_{n \times 1}\)` is a random vector with mean `\(\mu=E({\bf Y})\)` and covariance matrix `\(\Sigma=\text{Cov}({\bf Y})\)`. In addition, suppose `\({\bf A}_{r \times n}\)` is a matrix of constants and `\({\bf b}_{r \times 1}\)` is a vector of constants. Then `\begin{eqnarray*} E({\bf A Y} + {\bf b}) = {\bf A} E({\bf Y}) + {\bf b} = {\bf A} \mu + {\bf b} \end{eqnarray*}` and `\begin{eqnarray*} \text{Cov}({\bf A Y} + {\bf b}) = {\bf A} \text{Cov}({\bf Y}) {\bf A}' = {\bf A} \Sigma {\bf A}'. \end{eqnarray*}` --- ## Linear combinations Let `\({\bf W}_{r \times 1}\)` be a random vector with `\(E({\bf W})=\gamma\)`. Then `\begin{eqnarray*} \text{Cov}({\bf W},{\bf Y}) = E \left[ ({\bf W}-\gamma)({\bf Y}-\mu)' \right], \end{eqnarray*}` where `\(\text{Cov}({\bf W},{\bf Y})\)` is an `\((r \times n)\)` matrix of covariances with `\(ij^{th}\)` element equal to `\(\text{Cov}(W_i,Y_j)\)`. --- class: center, middle # What's next? ### Move on to the readings for the next module!