An explanation on variances I am having trouble understand what the following represent.
Let $\mathbb X^{p\times 1}$ denotes a column vector of random components $X_i,i=1(1)p.$ Then the expectation of $\mathbb x$ is defined as $$E(\mathbb X)=E\begin{pmatrix}X_1 \\ \vdots\\X_p\\\end{pmatrix}=\begin{pmatrix}EX_1 \\ \vdots\\EX_p\\\end{pmatrix}=\begin{pmatrix}\mu_1 \\ \vdots\\\mu_p\\\end{pmatrix}=\mathbb{\mu}
$$
Define $\text{Cov}(X_i,X_j)=E[(X_i-EX_i)(X_j-EX_j)^T]=\sigma_{ij}$. We extend the variance notation to the p-dimentional random vector $\mathbb X$ by the following matrix: 
$$E[(\mathbb X-E \mathbb X)(\mathbb X-E \mathbb X)^T] \\=E[(\mathbb X- \mathbb \mu)(\mathbb X-\mathbb \mu)^T] \\=E\left[\begin{pmatrix}X_1-\mu_1 \\ \vdots\\X_p-\mu_p\\\end{pmatrix} \begin{pmatrix}X_1-\mu_1&amp; \cdots &amp;X_p-\mu_p\\\end{pmatrix}\right]\\=\begin{pmatrix}\sigma_{11} &amp;\sigma_{12} &amp;\cdots &amp;\sigma_{1p}\\ \vdots &amp; \vdots &amp;\ddots &amp;\vdots \\\sigma_{p1} &amp;\sigma_{p2}&amp;\cdots&amp; \sigma_{pp}\\\end{pmatrix}=\Sigma$$ In similarly way co-variance between $X$ and $Y$ is defined.

That's all from me.
Let $\mathbb X^{p\times 1}$ denotes a column vector of random components $X_i,i=1(1)p.$ Then the expectation of $\mathbb x$ is defined as $$E(\mathbb X)=E\begin{pmatrix}X_1 \\ \vdots\\X_p\\\end{pmatrix}=\begin{pmatrix}EX_1 \\ \vdots\\EX_p\\\end{pmatrix}=\begin{pmatrix}\mu_1 \\ \vdots\\\mu_p\\\end{pmatrix}=\mathbb{\mu}
$$
Define $\text{Cov}(X_i,X_j)=E[(X_i-EX_i)(X_j-EX_j)^T]=\sigma_{ij}$. We extend the variance notation to the p-dimentional random vector $\mathbb X$ by the following matrix: 
$$E[(\mathbb X-E \mathbb X)(\mathbb X-E \mathbb X)^T] \\=E[(\mathbb X- \mathbb \mu)(\mathbb X-\mathbb \mu)^T] \\=E\left[\begin{pmatrix}X_1-\mu_1 \\ \vdots\\X_p-\mu_p\\\end{pmatrix} \begin{pmatrix}X_1-\mu_1&amp; \cdots &amp;X_p-\mu_p\\\end{pmatrix}\right]\\=\begin{pmatrix}\sigma_{11} &amp;\sigma_{12} &amp;\cdots &amp;\sigma_{1p}\\ \vdots &amp; \vdots &amp;\ddots &amp;\vdots \\\sigma_{p1} &amp;\sigma_{p2}&amp;\cdots&amp; \sigma_{pp}\\\end{pmatrix}=\Sigma$$ In similarly way co-variance between $X$ and $Y$ is defined.

That's all from me.