### 统计代写|蒙特卡洛方法代写monte carlo method代考| JOINTLY NORMAL RANDOM VARIABLES

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|蒙特卡洛方法代写monte carlo method代考|JOINTLY NORMAL RANDOM VARIABLES

It is helpful to view normally distributed random variables as simple transformations of standard normal – that is, $\mathrm{N}(0,1)$-distributed – random variables. In particular, let $X \sim \mathrm{N}(0,1)$. Then $X$ has density $f_{X}$ given by
$$f_{X}(x)=\frac{1}{\sqrt{2 \pi}} \mathrm{e}^{-\frac{x^{2}}{2}} .$$
Now consider the transformation $Z=\mu+\sigma X$. Then, by (1.15), $Z$ has density
$$f_{Z}(z)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} \mathrm{e}^{-\frac{(z-\mu)^{2}}{2 \sigma^{2}}} .$$
In other words, $Z \sim \mathrm{N}\left(\mu, \sigma^{2}\right)$. We can also state this as follows: if $Z \sim \mathrm{N}\left(\mu, \sigma^{2}\right)$, then $(Z-\mu) / \sigma \sim \mathrm{N}(0,1)$. This procedure is called standardization.

We now generalize this to $n$ dimensions. Let $X_{1}, \ldots, X_{n}$ be independent and standard normal random variables. The joint pdf of $\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)^{\top}$ is given by
$$f_{\mathbf{X}}(\mathbf{x})=(2 \pi)^{-n / 2} \mathrm{e}^{-\frac{1}{2} \mathbf{x}^{\top} \mathbf{x}}, \quad \mathbf{x} \in \mathbb{R}^{n} .$$
Consider the affine transformation (i.e., a linear transformation plus a constant vector)
$$\mathbf{Z}=\boldsymbol{\mu}+B \mathbf{X}$$
for some $m \times n$ matrix $B$. Note that, by Theorem 1.8.1, Z has expectation vector $\boldsymbol{\mu}$ and covariance matrix $\Sigma=B B^{\top}$. Any random vector of the form (1.23) is said to have a jointly normal or multivariate normal distribution. We write $\mathbf{Z} \sim N(\boldsymbol{\mu}, \Sigma)$. Suppose that $B$ is an invertible $n \times n$ matrix. Then, by (1.19), the density of $\mathbf{Y}=\mathbf{Z}-\boldsymbol{\mu}$ is given by
$$f_{\mathbf{Y}}(\mathbf{y})=\frac{1}{|B| \sqrt{(2 \pi)^{n}}} \mathrm{e}^{-\frac{1}{2}\left(B^{-1} \mathbf{y}\right)^{\top} B^{-1} \mathbf{y}}=\frac{1}{|B| \sqrt{(2 \pi)^{n}}} \mathrm{e}^{-\frac{1}{2} \mathbf{y}^{\top}\left(B^{-1}\right)^{\top} B^{-1} \mathbf{y}} .$$
We have $|B|=\sqrt{|\Sigma|}$ and $\left(B^{-1}\right)^{\top} B^{-1}=\left(B^{\top}\right)^{-1} B^{-1}=\left(B B^{\top}\right)^{-1}=\Sigma^{-1}$, so that
$$f_{\mathbf{Y}}(\mathbf{y})=\frac{1}{\sqrt{(2 \pi)^{n}|\Sigma|}} \mathrm{e}^{-\frac{1}{2} \mathbf{y}^{\top} \Sigma^{-1} \mathbf{y}} .$$
Because $\mathbf{Z}$ is obtained from $\mathbf{Y}$ by simply adding a constant vector $\boldsymbol{\mu}$, we have $f_{\mathbf{Z}}(\mathbf{z})=f_{\mathbf{Y}}(\mathbf{z}-\boldsymbol{\mu})$, and therefore
$$f_{\mathbf{Z}}(\mathbf{z})=\frac{1}{\sqrt{(2 \pi)^{n}|\Sigma|}} \mathrm{e}^{-\frac{1}{2}(\mathbf{z}-\boldsymbol{\mu})^{\top} \Sigma^{-1}(\mathbf{z}-\mu)}, \quad \mathbf{z} \in \mathbb{R}^{n} .$$
Note that this formula is very similar to that of the one-dimensional case.
Conversely, given a covariance matrix $\Sigma=\left(\sigma_{i j}\right)$, there exists a unique lower triangular matrix
$$B=\left(\begin{array}{cccc} b_{11} & 0 & \cdots & 0 \ b_{21} & b_{22} & \cdots & 0 \ \vdots & \vdots & & \vdots \ b_{n 1} & b_{n 2} & \cdots & b_{n n} \end{array}\right)$$
such that $\Sigma=B B^{\top}$. This matrix can be obtained efficiently via the Cholesky square root method; see Section A.1 of the Appendix.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|LIMIT THEOREMS

We briefly discuss two of the main results in probability: the law of large numbers and the central limit theorem. Both are associated with sums of independent random variables.

Let $X_{1}, X_{2}, \ldots$ be iid random variables with expectation $\mu$ and variance $\sigma^{2}$. For each $n$, let $S_{n}=X_{1}+\cdots+X_{n}$. Since $X_{1}, X_{2}, \ldots$ are iid, we have $\mathbb{E}\left[S_{n}\right]=n \mathbb{E}\left[X_{1}\right]=$ $n \mu$ and $\operatorname{Var}\left(S_{n}\right)=n \operatorname{Var}\left(X_{1}\right)=n \sigma^{2}$.

The law of large numbers states that $S_{n} / n$ is close to $\mu$ for large $n$. Here is the more precise statement.

Theorem 1.11.1 (Strong Law of Large Numbers) If $X_{1}, \ldots, X_{n}$ are iid with expectation $\mu$, then
$$\mathbb{P}\left(\lim {n \rightarrow \infty} \frac{S{n}}{n}=\mu\right)=1$$
The central limit theorem describes the limiting distribution of $S_{n}$ (or $S_{n} / n$ ), and it applies to both continuous and discrete random variables. Loosely, it states that the random sum $S_{n}$ has a distribution that is approximately normal, when $n$ is large. The more precise statement is given next.

Theorem 1.11.2 (Central Limit Theorem) If $X_{1}, \ldots, X_{n}$ are iid with expectation $\mu$ and variance $\sigma^{2}<\infty$, then for all $x \in \mathbb{R}$,
$$\lim {n \rightarrow \infty} \mathbb{P}\left(\frac{S{n}-n \mu}{\sigma \sqrt{n}} \leqslant x\right)=\Phi(x),$$
where $\Phi$ is the cdf of the standard normal distribution.
In other words, $S_{n}$ has a distribution that is approximately normal, with expectation $n \mu$ and variance $n \sigma^{2}$. To see the central limit theorem in action, consider Figure 1.2. The left part shows the pdfs of $S_{1}, \ldots, S_{4}$ for the case where the $\left{X_{i}\right}$ have a $U[0,1]$ distribution. The right part shows the same for the $\operatorname{Exp}(1)$ distribution. We clearly see convergence to a bell-shaped curve, characteristic of the normal distribution.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|POISSON PROCESSES

The Poisson process is used to model certain kinds of arrivals or patterns. Imagine, for example, a telescope that can detect individual photons from a faraway galaxy. The photons arrive at random times $T_{1}, T_{2}, \ldots .$ Let $N_{t}$ denote the number of arrivals in the time interval $[0, t]$, that is, $N_{t}=\sup \left{k: T_{k} \leqslant t\right}$. Note that the number of arrivals in an interval $I=(a, b]$ is given by $N_{b}-N_{a}$. We will also denote it by $N(a, b]$. A sample path of the arrival counting process $\left{N_{t}, t \geqslant 0\right}$ is given in Figure 1.3.

For this particular arrival process, one would assume that the number of arrivals in an interval $(a, b)$ is independent of the number of arrivals in interval $(c, d)$ when the two intervals do not intersect. Such considerations lead to the following definition:

Definition 1.12.1 (Poisson Process) An arrival counting process $N=\left{N_{t}\right}$ is called a Poisson process with rate $\lambda>0$ if
(a) The numbers of points in nonoverlapping intervals are independent.
(b) The number of points in interval $I$ has a Poisson distribution with mean $\lambda \times \operatorname{length}(I)$.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|JOINTLY NORMAL RANDOM VARIABLES

FX(X)=12圆周率和−X22.

F从(和)=12圆周率σ2和−(和−μ)22σ2.

FX(X)=(2圆周率)−n/2和−12X⊤X,X∈Rn.

F是(是)=1|乙|(2圆周率)n和−12(乙−1是)⊤乙−1是=1|乙|(2圆周率)n和−12是⊤(乙−1)⊤乙−1是.

F是(是)=1(2圆周率)n|Σ|和−12是⊤Σ−1是.

F从(和)=1(2圆周率)n|Σ|和−12(和−μ)⊤Σ−1(和−μ),和∈Rn.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|POISSON PROCESSES

(a) 非重叠区间中的点数是独立的。
(b) 区间点数一世具有均值的泊松分布λ×长度⁡(一世).

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