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# 数学代写|统计计算作业代写Statistical Computing代考|Multivariate normal distributions

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## 数学代写|统计计算作业代写Statistical Computing代考|important multivariate

One of the most important multivariate distributions is the multivariate normal distribution. In this section, we will derive the basic properties of the multivariate normal distribution and will discuss how to generate samples from this distribution.

Definition 2.1 Let $\mu \in \mathbb{R}^{d}$ be a vector and $\Sigma \in \mathbb{R}^{d \times d}$ be a symmetric, positive definite matrix. Then a random vector $X \in \mathbb{R}^{d}$ is normally distributed with mean $\mu$ and covariance matrix $\Sigma$, if the distribution of $X$ has density $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ given by
$$f(x)=\frac{1}{(2 \pi)^{d / 2}|\operatorname{det} \Sigma|^{1 / 2}} \exp \left(-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1}(x-\mu)\right)$$
for all $x \in \mathbb{R}^{d}$.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Using this interpretation

In this definition we consider the vector $x-\mu \in \mathbb{R}^{d}$ to be a $d \times 1$ matrix, and the expression $(x-\mu)^{\top}$ denotes the transpose of this vector, that is the vector $x-\mu$ interpreted as an $1 \times d$ matrix. Using this interpretation we have
$$(x-\mu)^{\top} \Sigma^{-1}(x-\mu)=\sum_{i, j=1}^{d}\left(x_{i}-\mu_{i}\right)\left(\Sigma^{-1}\right){i j}\left(x{j}-\mu_{j}\right)$$
The multivariate normal distribution from definition $2.1$ is a generalisation of the one-dimensional normal distribution: If $\Sigma$ is a diagonal matrix, say
$$\Sigma=\left(\begin{array}{cccc} \sigma_{1}^{2} & 0 & \ldots & 0 \ 0 & \sigma_{2}^{2} & \ldots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \ldots & \sigma_{d}^{2} \end{array}\right)$$
then $|\operatorname{det} \Sigma|=\prod_{i=1}^{d} \sigma_{i}^{2}$ and
$$\Sigma^{-1}=\left(\begin{array}{cccc} 1 / \sigma_{1}^{2} & 0 & \cdots & 0 \ 0 & 1 / \sigma_{2}^{2} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 / \sigma_{d}^{2} \end{array}\right)$$
and thus the density $f$ from (2.1) can be written as
\begin{aligned} f(x) &=\frac{1}{(2 \pi)^{d / 2}\left|\prod_{i=1}^{d} \sigma_{i}^{2}\right|^{1 / 2}} \exp \left(-\frac{1}{2} \sum_{i=1}^{d}\left(x_{i}-\mu_{i}\right) \frac{1}{\sigma_{i}^{2}}\left(x_{i}-\mu_{i}\right)\right) \ &=\prod_{i=1}^{d} \frac{1}{\left(2 \pi \sigma_{i}^{2}\right)^{1 / 2}} \exp \left(-\frac{\left(x_{i}-\mu_{i}\right)^{2}}{2 \sigma_{i}^{2}}\right) \ &=\prod_{i=1}^{d} f_{i}\left(x_{i}\right) \end{aligned}
where the function $f_{i}$, given by
$$f_{i}(x)=\frac{1}{\left(2 \pi \sigma_{i}^{2}\right)^{1 / 2}} \exp \left(-\frac{\left(x-\mu_{i}\right)^{2}}{2 \sigma_{i}^{2}}\right)$$
for all $x \in \mathbb{R}$, is the density of the one-dimensional normal distribution with mean $\mu_{i}$ and variance $\sigma_{i}^{2}$. This shows that $X$ is normally distributed on $\mathbb{R}^{d}$ with diagonal covariance matrix, if and only if the components $X_{i}$ for $i=1,2, \ldots, d$ are independent and normally distributed on $\mathbb{R}$.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|IMPORTANT MULTIVARIATE

F(X)=1(2圆周率)d/2|这⁡Σ|1/2经验⁡(−12(X−μ)⊤Σ−1(X−μ))

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|USING THIS INTERPRETATION

$$(x-\mu)^{\top} \Sigma^{-1}(x-\mu)=\sum_{i, j=1}^{d}\left(x_{i}-\mu_{i}\right)\left(\Sigma^{-1}\right){i j}\left(x{j}-\mu_{j}\right) .$$
The multivariate normal distribution from definition $2.1$ is a generalisation of the one-dimensional normal distribution: If $\Sigma$ is a diagonal matrix, say
$$\Sigma=\left(\begin{array}{cccc} \sigma_{1}^{2} & 0 & \ldots & 0 \ 0 & \sigma_{2}^{2} & \ldots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \ldots & \sigma_{d}^{2} \end{array}\right)$$
then $|\operatorname{det} \Sigma|=\prod_{i=1}^{d} \sigma_{i}^{2}$ and
$$\Sigma^{-1}=\left(\begin{array}{cccc} 1 / \sigma_{1}^{2} & 0 & \cdots & 0 \ 0 & 1 / \sigma_{2}^{2} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 / \sigma_{d}^{2} \end{array}\right)$$
and thus the density $f$ from (2.1) can be written as
\begin{aligned} f(x) &=\frac{1}{(2 \pi)^{d / 2}\left|\prod_{i=1}^{d} \sigma_{i}^{2}\right|^{1 / 2}} \exp \left(-\frac{1}{2} \sum_{i=1}^{d}\left(x_{i}-\mu_{i}\right) \frac{1}{\sigma_{i}^{2}}\left(x_{i}-\mu_{i}\right)\right) \ &=\prod_{i=1}^{d} \frac{1}{\left(2 \pi \sigma_{i}^{2}\right)^{1 / 2}} \exp \left(-\frac{\left(x_{i}-\mu_{i}\right)^{2}}{2 \sigma_{i}^{2}}\right) \ &=\prod_{i=1}^{d} f_{i}\left(x_{i}\right) \end{aligned}
where the function $f_{i}$, given by
$$f_{i}(x)=\frac{1}{\left(2 \pi \sigma_{i}^{2}\right)^{1 / 2}} \exp \left(-\frac{\left(x-\mu_{i}\right)^{2}}{2 \sigma_{i}^{2}}\right)$$