Ex.8 Multivariate Normal Distribution

$$ \begin{aligned} \int_{\mathbb{R}^m}f_X(\mathbf{x})d^mx &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\left|\boldsymbol{\Sigma}\right|^{1/2}}\exp\left(-\frac{1}{2}\left(\mathbf{x} - \boldsymbol{\mu}\right)^T\boldsymbol{\Sigma}^{-1}\left(\mathbf{x} - \boldsymbol{\mu}\right)\right)d^mx\\ &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}}\exp\left(-\frac{1}{2}\left(x_1^2 + \cdots + x_m^2\right)\right)d^mx\\ &=\prod_{k=1}^m\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}x_k^2\right)dx_k\\ &=1 \end{aligned} $$

$$I = \int_{-\infty}^{\infty}e^{-ax^2}dx = \int_{-\infty}^{\infty}e^{-ay^2}dy$$

$$\begin{aligned} I^2 &= \left(\int_{-\infty}^{\infty}e^{-ax^2}dx\right)^2\\ &= \left(\int_{-\infty}^{\infty}e^{-ax^2}dx\right)\left(\int_{-\infty}^{\infty}e^{-ay^2}dy\right)\\ &=\int_{\infty}^{\infty}dx\int_{-\infty}^{\infty}dye^{-a(x^2+y^2)}\\ &= \int_0^{\infty}rdr\int_0^{2\pi}d\theta e^{-ar^2}\\ &= 2\pi\left[-\frac{1}{2a}e^{-ar^2}\right]_0^{\infty}\\ &= \frac{\pi}{a}\\ \therefore I&= \sqrt{\frac{\pi}{a}} \end{aligned}$$

$$ \begin{aligned} \int_{\mathbb{R}^m}f_X(\mathbf{x})d^mx &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\left|\Sigma\right|^{1/2}}\exp\left(-\frac{1}{2}\left(\mathbf{x} - \boldsymbol{\mu}\right)^T\Sigma^{-1}\left(\mathbf{x} - \boldsymbol{\mu}\right)\right)d^mx\\ &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\left|O\Lambda O^T\right|^{1/2}}\exp\left(-\frac{1}{2}\left(\mathbf{x} - \boldsymbol{\mu}\right)^T\left(O\Lambda O^T\right)^{-1}\left(\mathbf{x} - \boldsymbol{\mu}\right)\right)d^mx\\ &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\left|\Lambda\right|^{1/2}}\exp\left(-\frac{1}{2}\left(O^{-1}\left(\mathbf{x} - \boldsymbol{\mu}\right)\right)^T\Lambda^{-1}\left(O^{-1}\left(\mathbf{x} - \boldsymbol{\mu}\right)\right)\right)d^mx\\ &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\left|\Lambda\right|^{1/2}}\exp\left(\sum_{k=1}^m-\frac{1}{2}\mathbf{o}_k^{-1T}(x_k-\mu_k)\lambda_k^{-1}\mathbf{o}_k^{-1}(x_k-\mu_k)\right)d^mx\\ &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\sum_{k=1}^m\lambda_k^{1/2}}\exp\left(\sum_{k=1}^m-\frac{1}{2}(x_k-\mu_k)\lambda_k^{-1}(x_k-\mu_k)\right)d^mx\\ &= \prod_{k=1}^m\int_{\mathbb{R}}\frac{1}{(2\pi)^{1/2}\lambda_k^{(1/2)}}\exp\left(-\frac{1}{2}\lambda_k^{-1}y_k^2\right)dy_k\\ &= \prod_{k=1}^m\frac{1}{(2\pi)^{1/2}\lambda_k^{(1/2)}}\sqrt{\frac{2\pi}{\lambda_k^{-1}}}\\ &= 1 \end{aligned} $$

$$ \begin{aligned} \mathbb{E}\left(X_k\right) &= \int_{\mathbb{R}^m}x_kf_X(\mathbf{x})d^mx\\ &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\sum_{k=1}^m\lambda_k^{1/2}}x_k\exp\left(\sum_{k=1}^m-\frac{1}{2}(x_k-\mu_k)\lambda_k^{-1}(x_k-\mu_k)\right)d^mx\\ &= \int_{\mathbb{R}^m}\frac{1}{\left(2\pi\right)^{m/2}\lambda_k^{1/2}}x_k\exp\left(-\frac{1}{2}\left(x_k-\mu_k\right)\lambda_k^{-1}\left(x_k-\mu_k\right)\right)dx_k\\ &=\frac{1}{\left(2\pi\right)^{m/2}\lambda_k^{1/2}}\int_{\mathbb{R}^m}\left(y_k + \mu_k\right)\exp\left(-\frac{1}{2}\lambda_k^{-1}y_k^2\right)dy_k\\ &= \frac{1}{\left(2\pi\right)^{m/2}\lambda_k^{1/2}}\left[-\frac{1}{\lambda_k^{-1}}\exp\left(-\frac{1}{2}\lambda_k^{-1}y_k^2\right)\right]_{-\infty}^{\infty} + \mu_k\\ &=\mu_k \end{aligned} $$