Chapter 56: multivariate normal distribution
Definition 37.1 probability density function (PDF) of normal distribution (= Gaussian distribution)
\[ \mathcal{N}\left(x \mid \mu, \sigma^2\right)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left\{-\frac{(x-\mu)^2}{2 \sigma^2}\right\} \]
If a continuous random variable \(X\) follows a normal distribution with mean 0 and variance \(\sigma^2\)
\[ \begin{aligned} X & \sim \mathrm{n}\left(\mu, \sigma^2\right)=\mathcal{N}\left(\mu, \sigma^2\right) \\ \Leftrightarrow f_X(x) & =\frac{1}{\sigma \sqrt{2 \pi}}\mathrm{e}^{\frac{-1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} =\frac{\mathrm{e}^{\frac{-1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}}{\sigma \sqrt{2 \pi}} =\mathrm{n}\left(x \mid \mu, \sigma^2\right) \\ & =\frac{1}{\sqrt{2 \pi} \sigma} \exp \left\{-\frac{(x-\mu)^2}{2 \sigma^2}\right\} =\mathcal{N}\left(x \mid \mu, \sigma^2\right) \end{aligned} \]
A random variable \(X\) can be standardized by subtracting the mean \(\mu\) and dividing by the standard deviation \(\sigma\), resulting in the standardized random variable \(Z\)
\[ Z=\dfrac{X-\mu}{\sigma}\text{ or } z=\dfrac{x-\mu}{\sigma} \]
The standardized random variable \(Z\) follows the standard normal distribution
\[ \begin{aligned} Z & \sim \mathrm{n}\left(0,1^2\right)=\mathcal{N}\left(0,1^2\right) \\ \Leftrightarrow f_Z(z) & =\frac{\mathrm{e}^{\frac{-1}{2}\left(\frac{x-0}{1}\right)^2}}{1 \cdot \sqrt{2 \pi}}=\frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{z^2}{2}\right\} \\ & =\mathrm{n}\left(z \mid 0,1^2\right)=\mathcal{N}\left(z \mid 0,1^2\right) \end{aligned} \]
To generalize from univariate random variables to multivariate random vectors, a random vector15
\[ \boldsymbol{Z}=\left\langle Z_{1},Z_{2},\cdots,Z_{p}\right\rangle =\begin{bmatrix}Z_{1} & Z_{2} & \cdots & Z_{p}\end{bmatrix}^{\intercal}=\begin{bmatrix}Z_{1}\\ Z_{2}\\ \vdots\\ Z_{p} \end{bmatrix} \]
with \(p\) random variable components is said to follow the standard multivariate normal distribution if and only if its joint PDF is given by
\[\begin{equation} \tag{56.1} f_{\boldsymbol{Z}}(\boldsymbol{z})=\frac{1}{(2\pi)^{p/2}}\exp\left\{ -\frac{\boldsymbol{z}^{\intercal}\boldsymbol{z}}{2}\right\} =\frac{1}{(2\pi)^{p/2}}\exp\left\{ -\frac{\boldsymbol{z}\cdot\boldsymbol{z}}{2}\right\} \end{equation}\]
(56.1) can be rewritten as the following
\[ \begin{aligned} f_{\boldsymbol{Z}}(\boldsymbol{z})= & \underbrace{\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{2\pi}}\cdots\frac{1}{\sqrt{2\pi}}}_{p\text{ times }}\exp\left\{ -\frac{z_{1}^{2}}{2}-\frac{z_{2}^{2}}{2}-\cdots-\frac{z_{p}^{2}}{2}\right\} \\ = & \frac{1}{\sqrt{2\pi}}\exp\left\{ -\frac{z_{1}^{2}}{2}\right\} \frac{1}{\sqrt{2\pi}}\exp\left\{ -\frac{z_{2}^{2}}{2}\right\} \cdots\frac{1}{\sqrt{2\pi}}\exp\left\{ -\frac{z_{p}^{2}}{2}\right\} \\ = & f\left(z_{1}\right)f\left(z_{2}\right)\cdots f\left(z_{p}\right) \end{aligned} \]
where
\[\begin{equation} \tag{56.2} f_{Z_{i}}\left(z_{i}\right)=\frac{1}{\sqrt{2\pi}}\exp\left\{ -\frac{z_{i}^{2}}{2}\right\} =f\left(z_{i}\right)\Rightarrow Z_{i}\sim\mathcal{N}\left(0,1^{2}\right)=\mathrm{n}\left(0,1^{2}\right) \end{equation}\]
\[ \begin{aligned} f_{Z_{i}}\left(z_{i}\right)= & \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}f_{\boldsymbol{Z}}\left(z_{1},\ldots,z_{i-1},z_{i},z_{i+1},\ldots,z_{p}\right)\mathrm{d}z_{1}\cdots\mathrm{d}z_{i-1}\mathrm{d}z_{i+1}\cdots\mathrm{d}z_{p}\\ = & \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}f\left(z_{1}\right)\cdots f\left(z_{i-1}\right)f\left(z_{i}\right)f\left(z_{i+1}\right)\cdots f\left(z_{p}\right)\mathrm{d}z_{1}\cdots\mathrm{d}z_{i-1}\mathrm{d}z_{i+1}\cdots\mathrm{d}z_{p}\\ = & f\left(z_{i}\right)\int_{-\infty}^{\infty}f\left(z_{1}\right)\mathrm{d}z_{1}\cdots\int_{-\infty}^{\infty}f\left(z_{i-1}\right)\mathrm{d}z_{i-1}\int_{-\infty}^{\infty}f\left(z_{i+1}\right)\mathrm{d}z_{i+1}\cdots\int_{-\infty}^{\infty}f\left(z_{p}\right)\mathrm{d}z_{p}\\ = & f\left(z_{i}\right) \overset{\eqref{eq:iidsnd}}{=}\frac{1}{\sqrt{2\pi}}\exp\left\{ -\frac{z_{i}^{2}}{2}\right\} \end{aligned} \]
Definition 15.1 covariance matrix of a random vector8
\[ \mathrm{C}\left[\boldsymbol{X}\right]=\mathrm{Cov}\left[\boldsymbol{X}\right]=\mathrm{V}\left[\boldsymbol{X}\right]=\mathrm{E}\left[\left[\boldsymbol{X}-\mathrm{E}\left(\boldsymbol{X}\right)\right]\left[\boldsymbol{X}-\mathrm{E}\left(\boldsymbol{X}\right)\right]^{\intercal}\right] \]
\[ \boldsymbol{X}=\left\langle X_{1},X_{2},\cdots,X_{p}\right\rangle =\begin{bmatrix}X_{1} & X_{2} & \cdots & X_{p}\end{bmatrix}^{\intercal}=\begin{bmatrix}X_{1}\\ X_{2}\\ \vdots\\ X_{p} \end{bmatrix} \]
\[ \mathrm{E}\left[\boldsymbol{X}\right]=\left\langle \mathrm{E}\left[X_{1}\right],\mathrm{E}\left[X_{2}\right],\cdots,\mathrm{E}\left[X_{p}\right]\right\rangle =\begin{bmatrix}\mathrm{E}\left[X_{1}\right] & \mathrm{E}\left[X_{2}\right] & \cdots & \mathrm{E}\left[X_{p}\right]\end{bmatrix}^{\intercal}=\begin{bmatrix}\mathrm{E}\left[X_{1}\right]\\ \mathrm{E}\left[X_{2}\right]\\ \vdots\\ \mathrm{E}\left[X_{p}\right] \end{bmatrix} \]
\[ \boldsymbol{X}-\mathrm{E}\left[\boldsymbol{X}\right]=\begin{bmatrix}X_{1}-\mathrm{E}\left[X_{1}\right]\\ X_{2}-\mathrm{E}\left[X_{2}\right]\\ \vdots\\ X_{p}-\mathrm{E}\left[X_{p}\right] \end{bmatrix} \]
\[ \begin{aligned} & \left[\boldsymbol{X}-\mathrm{E}\left(\boldsymbol{X}\right)\right]\left[\boldsymbol{X}-\mathrm{E}\left(\boldsymbol{X}\right)\right]^{\intercal}\\ = & \begin{bmatrix}X_{1}-\mathrm{E}\left[X_{1}\right]\\ X_{2}-\mathrm{E}\left[X_{2}\right]\\ \vdots\\ X_{p}-\mathrm{E}\left[X_{p}\right] \end{bmatrix}\begin{bmatrix}X_{1}-\mathrm{E}\left[X_{1}\right] & X_{2}-\mathrm{E}\left[X_{2}\right] & \cdots & X_{p}-\mathrm{E}\left[X_{p}\right]\end{bmatrix}\\ = & \begin{bmatrix}\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right) & \left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\left(X_{2}-\mathrm{E}\left[X_{2}\right]\right) & \cdots & \left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\\ \left(X_{2}-\mathrm{E}\left[X_{2}\right]\right)\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right) & \left(X_{2}-\mathrm{E}\left[X_{2}\right]\right)\left(X_{2}-\mathrm{E}\left[X_{2}\right]\right) & \cdots & \left(X_{2}-\mathrm{E}\left[X_{2}\right]\right)\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\\ \vdots & \vdots & \ddots & \vdots\\ \left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right) & \left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\left(X_{2}-\mathrm{E}\left[X_{2}\right]\right) & \cdots & \left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right) \end{bmatrix}\\ = & \begin{bmatrix}\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)^{2} & \cdots & \left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\\ \vdots & \ddots & \vdots\\ \left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right) & \cdots & \left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)^{2} \end{bmatrix} \end{aligned} \]
\[\begin{align} \tag{56.3} & \mathrm{E}\left[\left[\boldsymbol{X}-\mathrm{E}\left(\boldsymbol{X}\right)\right]\left[\boldsymbol{X}-\mathrm{E}\left(\boldsymbol{X}\right)\right]^{\intercal}\right]\\ = & \mathrm{E}\begin{bmatrix}\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)^{2} & \cdots & \left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\\ \vdots & \ddots & \vdots\\ \left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right) & \cdots & \left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)^{2} \end{bmatrix}\\ = & \begin{bmatrix}\mathrm{E}\left[\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)^{2}\right] & \cdots & \mathrm{E}\left[\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\right]\\ \vdots & \ddots & \vdots\\ \mathrm{E}\left[\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\right] & \cdots & \mathrm{E}\left[\left(X_{p}-\mathrm{E}\left[X_{p}\right]\right)^{2}\right] \end{bmatrix}\\ = & \begin{bmatrix}\mathrm{V}\left(X_{1},X_{1}\right) & \cdots & \mathrm{V}\left(X_{1},X_{p}\right)\\ \vdots & \ddots & \vdots\\ \mathrm{V}\left(X_{p},X_{1}\right) & \cdots & \mathrm{V}\left(X_{p},X_{p}\right) \end{bmatrix}=\begin{bmatrix}\mathrm{V}\left(X_{1},X_{1}\right) & \mathrm{V}\left(X_{1},X_{2}\right) & \cdots & \mathrm{V}\left(X_{1},X_{p}\right)\\ \mathrm{V}\left(X_{2},X_{1}\right) & \mathrm{V}\left(X_{2},X_{2}\right) & \cdots & \mathrm{V}\left(X_{2},X_{p}\right)\\ \vdots & \vdots & \ddots & \vdots\\ \mathrm{V}\left(X_{p},X_{1}\right) & \mathrm{V}\left(X_{p},X_{2}\right) & \cdots & \mathrm{V}\left(X_{p},X_{p}\right) \end{bmatrix}\\ = & \begin{bmatrix}\mathrm{V}\left(X_{1}\right) & \cdots & \mathrm{V}\left(X_{1},X_{p}\right)\\ \vdots & \ddots & \vdots\\ \mathrm{V}\left(X_{p},X_{1}\right) & \cdots & \mathrm{V}\left(X_{p}\right) \end{bmatrix}=\begin{bmatrix}\mathrm{V}\left(X_{1}\right) & \mathrm{V}\left(X_{1},X_{2}\right) & \cdots & \mathrm{V}\left(X_{1},X_{p}\right)\\ \mathrm{V}\left(X_{2},X_{1}\right) & \mathrm{V}\left(X_{2}\right) & \cdots & \mathrm{V}\left(X_{2},X_{p}\right)\\ \vdots & \vdots & \ddots & \vdots\\ \mathrm{V}\left(X_{p},X_{1}\right) & \mathrm{V}\left(X_{p},X_{2}\right) & \cdots & \mathrm{V}\left(X_{p}\right) \end{bmatrix}\\ = & \begin{bmatrix}\mathrm{V}\left(X_{1}\right) & \cdots & \mathrm{C}\left(X_{1},X_{p}\right)\\ \vdots & \ddots & \vdots\\ \mathrm{C}\left(X_{p},X_{1}\right) & \cdots & \mathrm{V}\left(X_{p}\right) \end{bmatrix}=\begin{bmatrix}\mathrm{V}\left(X_{1}\right) & \mathrm{C}\left(X_{1},X_{2}\right) & \cdots & \mathrm{C}\left(X_{1},X_{p}\right)\\ \mathrm{C}\left(X_{2},X_{1}\right) & \mathrm{V}\left(X_{2}\right) & \cdots & \mathrm{C}\left(X_{2},X_{p}\right)\\ \vdots & \vdots & \ddots & \vdots\\ \mathrm{C}\left(X_{p},X_{1}\right) & \mathrm{C}\left(X_{p},X_{2}\right) & \cdots & \mathrm{V}\left(X_{p}\right) \end{bmatrix}\\ = & \begin{bmatrix}\sigma_{1}^{2} & \cdots & \sigma_{1p}\\ \vdots & \ddots & \vdots\\ \sigma_{p1} & \cdots & \sigma_{p}^{2} \end{bmatrix}=\begin{bmatrix}\sigma_{1}^{2} & \sigma_{12} & \cdots & \sigma_{1p}\\ \sigma_{21} & \sigma_{2}^{2} & \cdots & \sigma_{2p}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{p1} & \sigma_{p2} & \cdots & \sigma_{p}^{2} \end{bmatrix}=\begin{bmatrix}\sigma_{11} & \sigma_{12} & \cdots & \sigma_{1p}\\ \sigma_{21} & \sigma_{22} & \cdots & \sigma_{2p}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{p1} & \sigma_{p2} & \cdots & \sigma_{pp} \end{bmatrix}=\left[\sigma_{ij}\right]_{p\times p}=\mathit{\Sigma} \end{align}\]
\[ \boldsymbol{X}\sim\mathcal{D}\left(\boldsymbol{\mu},\mathit{\Sigma}\right)=\mathrm{d}\left(\boldsymbol{\mu}_{\boldsymbol{X}},\mathit{\Sigma}_{\boldsymbol{X}}\right)=\mathrm{d}\left(\mathrm{E}\left[\boldsymbol{X}\right],\mathrm{C}\left[\boldsymbol{X}\right]\right)=\mathrm{d}\left(\mathrm{E}\left[\boldsymbol{X}\right],\mathrm{V}\left[\boldsymbol{X}\right]\right) \]
\[ \boldsymbol{Z}\sim\mathcal{N}\left(\boldsymbol{\mu}_{\boldsymbol{Z}},\mathit{\Sigma}_{\boldsymbol{Z}}\right)=\mathrm{n}\left(\mathrm{E}\left[\boldsymbol{Z}\right],\mathrm{V}\left[\boldsymbol{Z}\right]\right) \]
\[ \begin{aligned} & \mathrm{E}\left[\boldsymbol{Z}\right]=\begin{bmatrix}\mathrm{E}\left[Z_{1}\right]\\ \mathrm{E}\left[Z_{2}\right]\\ \vdots\\ \mathrm{E}\left[Z_{p}\right] \end{bmatrix}=\begin{bmatrix}\mathrm{E}\left[Z_{i}\right]\end{bmatrix}_{p\times1}\\ \Rightarrow & \mathrm{E}\left[Z_{i}\right]=\int_{-\infty}^{\infty}z_{i}f_{Z_{i}}\left(z_{i}\right)\mathrm{d}z_{i}\overset{\eqref{eq:iidsnd}}{=}\int_{-\infty}^{\infty}z_{i}\frac{\mathrm{e}^{\frac{-1}{2}z_{i}^{2}}}{\sqrt{2\pi}}\mathrm{d}z_{i}=0\\ \Rightarrow & \mathrm{E}\left[\boldsymbol{Z}\right]=\boldsymbol{0}\\ \Rightarrow & \boldsymbol{Z}\sim\mathcal{N}\left(\boldsymbol{\mu}_{\boldsymbol{Z}}=\boldsymbol{0},\mathit{\Sigma}_{\boldsymbol{Z}}\right)=\mathrm{n}\left(\boldsymbol{0},\mathrm{V}\left[\boldsymbol{Z}\right]\right) \end{aligned} \]
\[ \mathrm{V}\left(Z_{i}\right)=\int_{-\infty}^{\infty}\left(z_{i}-\mu_{Z_{i}}\right)^{2}f_{Z_{i}}\left(z_{i}\right)\mathrm{d}z_{i}\overset{\eqref{eq:iidsnd}}{=}\int_{-\infty}^{\infty}\left(z_{i}-0\right)^{2}\frac{\mathrm{e}^{\frac{-1}{2}z_{i}^{2}}}{\sqrt{2\pi}}\mathrm{d}z_{i}=1 \]
\[\begin{equation} \tag{56.4} \mathrm{V}\left(Z_{i},Z_{j}\right)\overset{i\ne j\Rightarrow Z_{i},Z_{j}\text{are independent}}{=}0 \end{equation}\]
\[ \begin{aligned} \mathrm{V}\left[\boldsymbol{Z}\right]= & \begin{bmatrix}\mathrm{V}\left(Z_{1}\right) & \mathrm{V}\left(Z_{1},Z_{2}\right) & \cdots & \mathrm{V}\left(Z_{1},Z_{p}\right)\\ \mathrm{V}\left(Z_{2},Z_{1}\right) & \mathrm{V}\left(Z_{2}\right) & \cdots & \mathrm{V}\left(Z_{2},Z_{p}\right)\\ \vdots & \vdots & \ddots & \vdots\\ \mathrm{V}\left(Z_{p},Z_{1}\right) & \mathrm{V}\left(Z_{p},Z_{2}\right) & \cdots & \mathrm{V}\left(Z_{p}\right) \end{bmatrix}=\begin{bmatrix}\sigma_{11} & \sigma_{12} & \cdots & \sigma_{1p}\\ \sigma_{21} & \sigma_{22} & \cdots & \sigma_{2p}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{p1} & \sigma_{p2} & \cdots & \sigma_{pp} \end{bmatrix}\\ = & \begin{bmatrix}\sigma_{1}^{2} & \sigma_{12} & \cdots & \sigma_{1p}\\ \sigma_{21} & \sigma_{2}^{2} & \cdots & \sigma_{2p}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{p1} & \sigma_{p2} & \cdots & \sigma_{p}^{2} \end{bmatrix}\overset{\ref{eq:idpsmnd}}{=}\begin{bmatrix}1 & 0 & \cdots & 0\\ 0 & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 \end{bmatrix}=I_{p\times p}=I_{p}=I \end{aligned} \]
\[ \boldsymbol{Z}\sim\mathcal{N}\left(\boldsymbol{\mu}_{\boldsymbol{Z}},\mathit{\Sigma}_{\boldsymbol{Z}}\right)=\mathrm{n}\left(\mathrm{E}\left[\boldsymbol{Z}\right],\mathrm{V}\left[\boldsymbol{Z}\right]\right)=\mathcal{N}\left(\boldsymbol{0},I\right)\Leftrightarrow\begin{cases} \boldsymbol{\mu}_{\boldsymbol{Z}}=\mathrm{E}\left[\boldsymbol{Z}\right]=\boldsymbol{0}=\left[0\right]_{p}=\left[0\right]_{p\times1}\\ \mathit{\Sigma}_{\boldsymbol{Z}}=\mathrm{V}\left[\boldsymbol{Z}\right]=I=I_{p}=I_{p\times p} \end{cases} \]
\[ \begin{aligned} \boldsymbol{Z}= & \begin{bmatrix}Z_{1}\\ Z_{2}\\ \vdots\\ Z_{p} \end{bmatrix}=\begin{bmatrix}\dfrac{X_{1}-\mu_{1}}{\sigma_{1}}\\ \dfrac{X_{2}-\mu_{2}}{\sigma_{2}}\\ \vdots\\ \dfrac{X_{p}-\mu_{p}}{\sigma_{p}} \end{bmatrix}=\begin{bmatrix}\dfrac{1}{\sigma_{1}} & 0 & \cdots & 0\\ 0 & \dfrac{1}{\sigma_{2}} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \dfrac{1}{\sigma_{p}} \end{bmatrix}\begin{bmatrix}X_{1}-\mu_{1}\\ X_{2}-\mu_{2}\\ \vdots\\ X_{p}-\mu_{p} \end{bmatrix}\\ = & \begin{bmatrix}\dfrac{1}{\sigma_{1}} & 0 & \cdots & 0\\ 0 & \dfrac{1}{\sigma_{2}} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \dfrac{1}{\sigma_{p}} \end{bmatrix}\left(\begin{bmatrix}X_{1}\\ X_{2}\\ \vdots\\ X_{p} \end{bmatrix}-\begin{bmatrix}\mu_{1}\\ \mu_{2}\\ \vdots\\ \mu_{p} \end{bmatrix}\right)=B^{-1}\left(\boldsymbol{X}-\boldsymbol{\mu}\right)\\ \Rightarrow\boldsymbol{X}= & B\boldsymbol{Z}+\boldsymbol{\mu} \end{aligned} \]
\[ \boldsymbol{X} = B\boldsymbol{Z} + \boldsymbol{\mu} = T\left(\boldsymbol{Z}\right) \]
\[\begin{equation} \tag{56.5} \mathit{\Sigma}=\mathit{\Sigma}_{\boldsymbol{X}}=\mathrm{V}\left[\boldsymbol{X}\right]=\mathrm{V}\left[B\boldsymbol{Z}+\boldsymbol{\mu}\right]=B\mathrm{V}\left[\boldsymbol{Z}\right]B^{\intercal}=BIB^{\intercal}=BB^{\intercal} \end{equation}\]
Consider two infinitesimal volumes of \(p\)-dimensional parallelepipeds in the different \(\mathbb{R}^p\) spaces16
\[ V_{\boldsymbol{x}}=\left[x_{1},x_{1}+\mathrm{d}x_{1}\right]\times\left[x_{2},x_{2}+\mathrm{d}x_{2}\right]\times\cdots\times\left[x_{p},x_{p}+\mathrm{d}x_{p}\right] \]
and
\[ V_{\boldsymbol{z}}=\left[z_{1},z_{1}+\mathrm{d}z_{1}\right]\times\left[z_{2},z_{2}+\mathrm{d}z_{2}\right]\times\cdots\times\left[z_{p},z_{p}+\mathrm{d}z_{p}\right] \]
Their relationship under linear transformation is
\[ \begin{aligned} V_{\boldsymbol{x}}=T\left(V_{\boldsymbol{z}}\right)= & \left[T\left(z_{1}\right),T\left(z_{1}\right)+T\left(\mathrm{d}z_{1}\right)\right]\\ & \times\left[T\left(z_{2}\right),T\left(z_{2}\right)+T\left(\mathrm{d}z_{2}\right)\right]\\ & \times\cdots\\ & \times\left[T\left(z_{p}\right),T\left(z_{p}\right)+T\left(\mathrm{d}z_{p}\right)\right] \end{aligned} \]
and
\[ \mathrm{d}x_{i}=\sum_{j}\dfrac{\partial x_{i}}{\partial z_{j}}\mathrm{d}z_{j} \]
For examples in 2 dimension,
\[ \begin{bmatrix}\mathrm{d}x_{1}\\ \mathrm{d}x_{2} \end{bmatrix}=\begin{bmatrix}\dfrac{\partial x_{1}}{\partial z_{1}} & \dfrac{\partial x_{1}}{\partial z_{2}}\\ \dfrac{\partial x_{2}}{\partial z_{1}} & \dfrac{\partial x_{2}}{\partial z_{2}} \end{bmatrix}\begin{bmatrix}\mathrm{d}z_{1}\\ \mathrm{d}z_{2} \end{bmatrix} \] Two element infinitesimal one-directional vectors of \(\boldsymbol{Z}\) transformed into another space of \(\boldsymbol{X}\) are
\[ T\left(\mathrm{d}\boldsymbol{z}_{1}\right)=\begin{bmatrix}\dfrac{\partial x_{1}}{\partial z_{1}} & \dfrac{\partial x_{1}}{\partial z_{2}}\\ \dfrac{\partial x_{2}}{\partial z_{1}} & \dfrac{\partial x_{2}}{\partial z_{2}} \end{bmatrix}\begin{bmatrix}\mathrm{d}z_{1}\\ 0 \end{bmatrix}=\begin{bmatrix}\dfrac{\partial x_{1}}{\partial z_{1}}\mathrm{d}z_{1}\\ \dfrac{\partial x_{2}}{\partial z_{1}}\mathrm{d}z_{1} \end{bmatrix} \]
and
\[ T\left(\mathrm{d}\boldsymbol{z}_{2}\right)=\begin{bmatrix}\dfrac{\partial x_{1}}{\partial z_{1}} & \dfrac{\partial x_{1}}{\partial z_{2}}\\ \dfrac{\partial x_{2}}{\partial z_{1}} & \dfrac{\partial x_{2}}{\partial z_{2}} \end{bmatrix}\begin{bmatrix}0\\ \mathrm{d}z_{2} \end{bmatrix}=\begin{bmatrix}\dfrac{\partial x_{1}}{\partial z_{2}}\mathrm{d}z_{2}\\ \dfrac{\partial x_{2}}{\partial z_{2}}\mathrm{d}z_{2} \end{bmatrix} \]
Their corresponding area(volume) in the space of \(\boldsymbol{X}\) is
\[ \begin{aligned} & \int_{A_{\boldsymbol{x}}}\mathrm{d}A_{\boldsymbol{x}}=\int_{A_{\boldsymbol{x}}}\mathrm{d}x_{1}\mathrm{d}x_{2}=\int_{T\left(A_{\boldsymbol{z}}\right)}\mathrm{d}x_{1}\mathrm{d}x_{2}\\ = & \int_{A_{\boldsymbol{z}}}\left|\begin{bmatrix}T\left(\mathrm{d}\boldsymbol{z}_{1}\right) & T\left(\mathrm{d}\boldsymbol{z}_{2}\right)\end{bmatrix}\right|=\int_{A_{\boldsymbol{z}}}\begin{vmatrix}\dfrac{\partial x_{1}}{\partial z_{1}}\mathrm{d}z_{1} & \dfrac{\partial x_{1}}{\partial z_{2}}\mathrm{d}z_{2}\\ \dfrac{\partial x_{2}}{\partial z_{1}}\mathrm{d}z_{1} & \dfrac{\partial x_{2}}{\partial z_{2}}\mathrm{d}z_{2} \end{vmatrix}\\ = & \int_{A_{\boldsymbol{z}}}\begin{vmatrix}\dfrac{\partial x_{1}}{\partial z_{1}} & \dfrac{\partial x_{1}}{\partial z_{2}}\\ \dfrac{\partial x_{2}}{\partial z_{1}} & \dfrac{\partial x_{2}}{\partial z_{2}} \end{vmatrix}\mathrm{d}z_{1}\mathrm{d}z_{2}=\int_{A_{\boldsymbol{z}}}\left|J\right|\mathrm{d}A_{\boldsymbol{z}} \end{aligned} \]
To generalize for volumes in \(p\) dimension,
\[ \begin{aligned} & \int_{V_{\boldsymbol{x}}}\mathrm{d}V_{\boldsymbol{x}}=\int_{V_{\boldsymbol{x}}}\mathrm{d}x_{1}\mathrm{d}x_{2}\cdots\mathrm{d}x_{p}=\int_{T\left(V_{\boldsymbol{z}}\right)}\mathrm{d}x_{1}\mathrm{d}x_{2}\cdots\mathrm{d}x_{p}\\ = & \int_{A_{\boldsymbol{z}}}\left|\begin{bmatrix}T\left(\mathrm{d}\boldsymbol{z}_{1}\right) & T\left(\mathrm{d}\boldsymbol{z}_{2}\right) & \cdots & T\left(\mathrm{d}\boldsymbol{z}_{p}\right)\end{bmatrix}\right|=\int_{V_{\boldsymbol{z}}}\begin{vmatrix}\left[\dfrac{\partial x_{i}}{\partial z_{j}}\mathrm{d}z_{j}\right]_{p\times p}\end{vmatrix}\\ = & \int_{V_{\boldsymbol{z}}}\begin{vmatrix}\left[\dfrac{\partial x_{i}}{\partial z_{j}}\right]_{p\times p}\end{vmatrix}\mathrm{d}z_{1}\mathrm{d}z_{2}\cdots\mathrm{d}z_{p}=\int_{V_{\boldsymbol{z}}}\left|J\right|\mathrm{d}V_{\boldsymbol{z}} \end{aligned} \]
i.e.
\[\begin{equation} \tag{56.6} \int_{V_{\boldsymbol{x}}}\mathrm{d}V_{\boldsymbol{x}}=\int_{V_{\boldsymbol{z}}}\left|J\right|\mathrm{d}V_{\boldsymbol{z}} \end{equation}\]
where \(J\) is a Jacobian matrix
\[ J=\left[\dfrac{\partial x_{i}}{\partial z_{j}}\right]_{p\times p}=\dfrac{\partial\boldsymbol{x}}{\partial\boldsymbol{z}} \]
or \(\left|J\right|\) is a Jacobian determinant or simply Jacobian
\[ \left|J\right|=\left|\dfrac{\partial x_{i}}{\partial z_{j}}\right|_{p\times p}=\left|\dfrac{\partial\boldsymbol{x}}{\partial\boldsymbol{z}}\right| \]
The probability of the same event should be invariant under transformation.
\[ \begin{aligned} & \int_{V_{\boldsymbol{x}}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\mathrm{d}V_{\boldsymbol{x}}=\int_{V_{\boldsymbol{x}}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\mathrm{d}x_{1}\mathrm{d}x_{2}\cdots\mathrm{d}x_{p}\\ = & \int_{T\left(V_{\boldsymbol{z}}\right)}f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\mathrm{d}x_{1}\mathrm{d}x_{2}\cdots\mathrm{d}x_{p}\\ = & \int_{V_{\boldsymbol{z}}}f_{\boldsymbol{Z}}\left(\boldsymbol{z}\right)\mathrm{d}V_{\boldsymbol{z}}=\int_{V_{\boldsymbol{z}}}f_{\boldsymbol{Z}}\left(\boldsymbol{z}\right)\mathrm{d}z_{1}\mathrm{d}z_{2}\cdots\mathrm{d}z_{p} \end{aligned} \]
i.e.
\[\begin{equation} \tag{56.7} \int_{V_{\boldsymbol{x}}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\mathrm{d}V_{\boldsymbol{x}}=\int_{V_{\boldsymbol{z}}}f_{\boldsymbol{Z}}\left(\boldsymbol{z}\right)\mathrm{d}V_{\boldsymbol{z}} \end{equation}\]
\[ \begin{cases} \int_{V_{\boldsymbol{x}}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\mathrm{d}V_{\boldsymbol{x}}=\int_{V_{\boldsymbol{z}}}f_{\boldsymbol{Z}}\left(\boldsymbol{z}\right)\mathrm{d}V_{\boldsymbol{z}} & \ref{eq:ipmit}\\ \int_{V_{\boldsymbol{x}}}\mathrm{d}V_{\boldsymbol{x}}=\int_{V_{\boldsymbol{z}}}\left|J\right|\mathrm{d}V_{\boldsymbol{z}} & \ref{eq:mitj} \end{cases} \]
\[\begin{align} \boldsymbol{Z}= & B^{-1}\left(\boldsymbol{X}-\boldsymbol{\mu}\right)\nonumber \\ \boldsymbol{z}= & B^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right) \tag{56.8}\\ \boldsymbol{X}= & B\boldsymbol{Z}+\boldsymbol{\mu}\nonumber \\ \boldsymbol{x}= & B\boldsymbol{z}+\boldsymbol{\mu}\nonumber \\ J=\left[\dfrac{\partial x_{i}}{\partial z_{j}}\right]_{p\times p}=\dfrac{\partial\boldsymbol{x}}{\partial\boldsymbol{z}}= & B \tag{56.9}\\ \left|J\right|=\left|\dfrac{\partial x_{i}}{\partial z_{j}}\right|_{p\times p}=\left|\dfrac{\partial\boldsymbol{x}}{\partial\boldsymbol{z}}\right|= & \left|B\right|\nonumber \end{align}\]
\[ \begin{aligned} \int_{V_{\boldsymbol{z}}}f_{\boldsymbol{Z}}\left(\boldsymbol{z}\right)\mathrm{d}V_{\boldsymbol{z}}\overset{\ref{eq:ipmit}}{=} & \int_{V_{\boldsymbol{x}}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\mathrm{d}V_{\boldsymbol{x}}=\int_{V_{\boldsymbol{x}}}\mathrm{d}V_{\boldsymbol{x}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\\ \overset{\ref{eq:mitj}}{=}\int_{V_{\boldsymbol{z}}}\left|J\right|\mathrm{d}V_{\boldsymbol{z}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\left(\boldsymbol{z}\right)\right)= & \int_{V_{\boldsymbol{z}}}f_{\boldsymbol{X}}\left(\boldsymbol{x}\left(\boldsymbol{z}\right)\right)\left|J\right|\mathrm{d}V_{\boldsymbol{z}}\\ f_{\boldsymbol{Z}}\left(\boldsymbol{z}\right)\overset{\Downarrow}{=} & f_{\boldsymbol{X}}\left(\boldsymbol{x}\left(\boldsymbol{z}\right)\right)\left|J\right|\\ f_{\boldsymbol{X}}\left(\boldsymbol{x}\left(\boldsymbol{z}\right)\right)\overset{\Downarrow}{=} & \left|J\right|^{-1}f_{\boldsymbol{Z}}\left(\boldsymbol{z}\right)\overset{\ref{eq:smnd}}{=}\left|J\right|^{-1}\frac{1}{(2\pi)^{p/2}}\exp\left\{ \frac{-\boldsymbol{z}^{\intercal}\boldsymbol{z}}{2}\right\} \\ f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\overset{\Downarrow}{=} & \left|J\right|^{-1}f_{\boldsymbol{Z}}\left(\boldsymbol{z}\left(\boldsymbol{x}\right)\right)\overset{\ref{eq:sltj},\ref{eq:sltx2z}}{=}\left|B\right|^{-1}f_{\boldsymbol{Z}}\left(B^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right)\\ = & \left|B\right|^{-1}(2\pi)^{-p/2}\exp\left\{ \frac{-1}{2}\left[B^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right]^{\intercal}\left[B^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right]\right\} \\ = & \left|B\right|^{-1/2}\left|B\right|^{-1/2}(2\pi)^{-p/2}\exp\left\{ \frac{-1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^{\intercal}\left(B^{-1}\right)^{\intercal}B^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\} \\ = & \left|B\right|^{-1/2}\left|B^{\intercal}\right|^{-1/2}(2\pi)^{-p/2}\exp\left\{ \frac{-1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^{\intercal}\left(B^{\intercal}\right)^{-1}B^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\} \\ = & \left|BB^{\intercal}\right|^{-1/2}(2\pi)^{-p/2}\exp\left\{ \frac{-1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^{\intercal}\left(BB^{\intercal}\right)^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\} \\ \overset{\ref{eq:covlt}}{=} & \left|\mathit{\Sigma}\right|^{-1/2}(2\pi)^{-p/2}\exp\left\{ \frac{-1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^{\intercal}\mathit{\Sigma}^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\} \\ = & \left(\left|\mathit{\Sigma}\right|\left(2\pi\right)^{p}\right)^{-1/2}\exp\left\{ \frac{-1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^{\intercal}\mathit{\Sigma}^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\} \end{aligned} \]
Definition 15.2 probability density function (PDF) of multivariate normal distribution (= multivariate Gaussian distribution)
\[ \mathcal{N}\left(\boldsymbol{x}\mid\boldsymbol{\mu},\mathit{\Sigma}\right)=f_{\boldsymbol{X}}\left(\boldsymbol{x}\right)=\left(\left|\mathit{\Sigma}\right|\left(2\pi\right)^{p}\right)^{-1/2}\exp\left\{ \frac{-1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^{\intercal}\mathit{\Sigma}^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\} \]
Definition 15.3 correlation coefficient
\[ \rho_{ij}=\dfrac{\sigma_{ij}}{\sqrt{\sigma_{ii}}\sqrt{\sigma_{jj}}}=\dfrac{\sigma_{ij}}{\sqrt{\sigma_{i}^{2}}\sqrt{\sigma_{j}^{2}}}=\dfrac{\sigma_{ij}}{\sigma_{i}\sigma_{j}}=\dfrac{\mathrm{V}\left(X_{i},X_{j}\right)}{\sqrt{\mathrm{V}\left(X_{i}\right)}\sqrt{\mathrm{V}\left(X_{j}\right)}}=\mathrm{R}\left(X_{i},X_{j}\right) \]
56.1 bivariate normal distribution
\(p=2\) is the case of bivariate normal distribution
\[ \mathit{\Sigma}=\left[\sigma_{ij}\right]_{2\times2}=\begin{bmatrix}\sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22} \end{bmatrix}=\begin{bmatrix}\sigma_{1}^{2} & \sigma_{12}\\ \sigma_{21} & \sigma_{2}^{2} \end{bmatrix}=\begin{bmatrix}\sigma_{1}^{2} & \sigma_{1}\sigma_{2}\rho_{12}\\ \sigma_{2}\sigma_{1}\rho_{21} & \sigma_{2}^{2} \end{bmatrix}=\begin{bmatrix}\sigma_{1}^{2} & \sigma_{1}\sigma_{2}\rho\\ \sigma_{2}\sigma_{1}\rho & \sigma_{2}^{2} \end{bmatrix} \]
\[ \rho_{12}=\rho=\rho_{21} \]
\[ \left|\mathit{\Sigma}\right|=\begin{vmatrix}\sigma_{1}^{2} & \sigma_{1}\sigma_{2}\rho_{12}\\ \sigma_{2}\sigma_{1}\rho_{21} & \sigma_{2}^{2} \end{vmatrix}=\sigma_{1}^{2}\sigma_{2}^{2}\left(1-\rho_{12}\rho_{21}\right)=\sigma_{1}^{2}\sigma_{2}^{2}\left(1-\rho^{2}\right) \]
\[ \left[\begin{array}{ll} a & b\\ c & d \end{array}\right]^{-1}=\frac{1}{\begin{vmatrix}a & b\\ c & d \end{vmatrix}}\left[\begin{array}{cc} d & -b\\ -c & a \end{array}\right] \]
\[ \begin{aligned} \mathit{\Sigma}^{-1}= & \frac{1}{\left|\mathit{\Sigma}\right|}\left[\begin{array}{cc} \sigma_{2}^{2} & -\sigma_{1}\sigma_{2}\rho\\ -\sigma_{2}\sigma_{1}\rho & \sigma_{1}^{2} \end{array}\right]\\ = & \frac{1}{\sigma_{1}^{2}\sigma_{2}^{2}\left(1-\rho^{2}\right)}\left[\begin{array}{cc} \sigma_{2}^{2} & -\sigma_{1}\sigma_{2}\rho\\ -\sigma_{2}\sigma_{1}\rho & \sigma_{1}^{2} \end{array}\right]\\ = & \frac{1}{\left(1-\rho^{2}\right)}\left[\begin{array}{cc} \dfrac{1}{\sigma_{1}^{2}} & \dfrac{-\rho}{\sigma_{1}\sigma_{2}}\\ \dfrac{-\rho}{\sigma_{2}\sigma_{1}} & \dfrac{1}{\sigma_{2}^{2}} \end{array}\right] \end{aligned} \]
\[ \begin{aligned} & \mathcal{N}\left(\boldsymbol{x}=\begin{bmatrix}x_{1}\\ x_{2} \end{bmatrix} \middle| \boldsymbol{\mu}=\begin{bmatrix}\mu_{1}\\ \mu_{2} \end{bmatrix},\mathit{\Sigma}=\begin{bmatrix}\sigma_{1}^{2} & \sigma_{1}\sigma_{2}\rho\\ \sigma_{2}\sigma_{1}\rho & \sigma_{2}^{2} \end{bmatrix}\right)\\ = & \left(\left|\mathit{\Sigma}\right|\left(2\pi\right)^{p=2}\right)^{-1/2}\exp\left\{ \frac{-1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^{\intercal}\mathit{\Sigma}^{-1}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\} \\ = & \left(\sigma_{1}^{2}\sigma_{2}^{2}\left(1-\rho^{2}\right)\left(2\pi\right)^{2}\right)^{-1/2}\exp\left\{ \frac{-1}{2}\begin{bmatrix}x_{1}-\mu_{1}\\ x_{2}-\mu_{2} \end{bmatrix}^{\intercal}\mathit{\Sigma}^{-1}\begin{bmatrix}x_{1}-\mu_{1}\\ x_{2}-\mu_{2} \end{bmatrix}\right\} \\ = & \dfrac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left\{ \frac{-1}{2}\begin{bmatrix}x_{1}-\mu_{1}\\ x_{2}-\mu_{2} \end{bmatrix}^{\intercal}\frac{1}{\left(1-\rho^{2}\right)}\left[\begin{array}{cc} \dfrac{1}{\sigma_{1}^{2}} & \dfrac{-\rho}{\sigma_{1}\sigma_{2}}\\ \dfrac{-\rho}{\sigma_{2}\sigma_{1}} & \dfrac{1}{\sigma_{2}^{2}} \end{array}\right]\begin{bmatrix}x_{1}-\mu_{1}\\ x_{2}-\mu_{2} \end{bmatrix}\right\} \\ = & \dfrac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left\{ \frac{-1}{2\left(1-\rho^{2}\right)}\begin{bmatrix}x_{1}-\mu_{1}\\ x_{2}-\mu_{2} \end{bmatrix}^{\intercal}\left[\begin{array}{cc} \dfrac{1}{\sigma_{1}^{2}} & \dfrac{-\rho}{\sigma_{1}\sigma_{2}}\\ \dfrac{-\rho}{\sigma_{2}\sigma_{1}} & \dfrac{1}{\sigma_{2}^{2}} \end{array}\right]\begin{bmatrix}x_{1}-\mu_{1}\\ x_{2}-\mu_{2} \end{bmatrix}\right\} \\ = & \dfrac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left\{ \frac{-1}{2\left(1-\rho^{2}\right)}\left[\left(\dfrac{x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}-2\rho\left(\dfrac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\left(\dfrac{x_{2}-\mu_{2}}{\sigma_{2}}\right)+\left(\dfrac{x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\} \end{aligned} \]
Definition 15.4 probability density function (PDF) of bivariate normal distribution (= bivariate Gaussian distribution)
\[ \begin{aligned} & \mathcal{N}\left(\begin{bmatrix}x_{1}\\ x_{2} \end{bmatrix} \middle| \begin{bmatrix}\mu_{1}\\ \mu_{2} \end{bmatrix},\begin{bmatrix}\sigma_{1}^{2} & \sigma_{1}\sigma_{2}\rho\\ \sigma_{2}\sigma_{1}\rho & \sigma_{2}^{2} \end{bmatrix}\right)\\ = & \dfrac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left\{ \frac{-1}{2\left(1-\rho^{2}\right)}\left[\left(\dfrac{x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}-2\rho\left(\dfrac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\left(\dfrac{x_{2}-\mu_{2}}{\sigma_{2}}\right)+\left(\dfrac{x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\} \end{aligned} \]