### 统计代写|金融统计代写Mathematics with Statistics for Finance代考|CORRELATION

statistics-lab™ 为您的留学生涯保驾护航 在代写金融统计Mathematics with Statistics for Finance G1GH方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融统计Mathematics with Statistics for Finance G1GH方面经验极为丰富，各种代写金融统计Mathematics with Statistics for Finance G1GH相关的作业也就用不着说。

• 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|金融统计代写Mathematics with Statistics for Finance代考|CORRELATION

Closely related to the concept of covariance is correlation. To get the correlation of two variables, we simply divide their covariance by their respective standard deviations:
$$\rho_{X Y}=\frac{\sigma_{X Y}}{\sigma_{X} \sigma_{Y}}$$
Correlation has the nice property that it varies between $-1$ and $+1$. If two variables have a correlation of $+1$, then we say they are perfectly correlated. If the ratio of one variable to another is always the same and positive then the two variables will be perfectly correlated.

If two variables are highly correlated, it is often the case that one variable causes the other variable, or that both variables share a common underlying driver. We will see in later chapters, though, that it is very easy for two random variables with no causal link to be highly correlated. Correlation does not prove causation. Similarly, if two variables are uncorrelated, it does not necessarily follow that they are unrelated. For example, a random variable that is symmetrical around zero and the square of that variable will have zero correlation.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|PORTFOLIO VARIANCE AND HEDGING

If we have a portfolio of securities and we wish to determine the variance of that portfolio, all we need to know is the variance of the underlying securities and their respective correlations.

For example, if we have two securities with random returns $X_{A}$ and $X_{B}$, with means $\mu_{A}$ and $\mu_{B}$ and standard deviations $\sigma_{A}$ and $\sigma_{B}$, respectively, we can calculate the variance of $X_{A}$ plus $X_{B}$ as follows:
$$\sigma_{A+B}^{2}=\sigma_{A}^{2}+\sigma_{B}^{2}+2 \rho_{A B} \sigma_{A} \sigma_{B}$$
where $\rho_{A B}$ is the correlation between $X_{A}$ and $X_{B}$. The proof is left as an exercise. Notice that the last term can either increase or decrease the total variance. Both standard deviations must be positive; therefore, if the correlation is positive, the overall variance will be higher compared to the case where the correlation is negative.

If the variance of both securities is equal, then Equation $3.29$ simplifies to:
$$\sigma_{A+B}^{2}=2 \sigma^{2}\left(1+\rho_{A B}\right) \text { where } \sigma_{A}^{2}=\sigma_{B}^{2}=\sigma^{2}$$
Now we know that the correlation can vary between $-1$ and $+1$, so, substituting into our new equation, the portfolio variance must be bound by 0 and $4 \sigma^{2}$. If we take the square root of both sides of the equation, we see that the standard deviation is bound by 0 and $2 \sigma$. Intuitively this should make

sense. If, on the one hand, we own one share of an equity with a standard deviation of $\$ 10$and then purchase another share of the same equity, then the standard deviation of our two-share portfolio must be$\$20$ (trivially, the correlation of a random variable with itself must be one). On the other hand, if we own one share of this equity and then purchase another security that always generates the exact opposite return, the portfolio is perfectly balanced. The returns are always zero, which implies a standard deviation of zero.

In the special case where the correlation between the two securities is zero, we can further simplify our equation. For the standard deviation:
$$\rho_{A B}=0 \Rightarrow \sigma_{A+B}=\sqrt{2} \sigma$$
We can extend Equation $3.29$ to any number of variables:
\begin{aligned} Y &=\sum_{i=1}^{n} X_{i} \ \sigma_{Y}^{2} &=\sum_{i=1}^{n} \sum_{j=1}^{n} \rho_{i j} \sigma_{i} \sigma_{j} \end{aligned}
In the case where all of the $X_{i}$ ‘s are uncorrelated and all the variances are equal to $\sigma$, Equation $3.32$ simplifies to:
$$\sigma_{Y}=\sqrt{n} \sigma \quad \text { iff } \rho_{i j}=0 \forall i \neq j$$

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|MOMENTS

Previously, we defined the mean of a variable $X$ as:
$$\mu=E[X]$$
It turns out that we can generalize this concept as follows:
$$m_{k}=E\left[X^{k}\right]$$
We refer to $m_{k}$ as the $k$ th moment of $X$. The mean of $X$ is also the first moment of $X$.
Similarly, we can generalize the concept of variance as follows:
$$\mu_{k}=E\left[(X-\mu)^{k}\right]$$
We refer to $\mu_{k}$ as the $k$ th central moment of $X$. We say that the moment is central because it is central around the mean. Variance is simply the second central moment.

While we can easily calculate any central moment, in risk management it is very rare that we are interested in anything beyond the fourth central moment.

ρX是=σX是σXσ是

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|PORTFOLIO VARIANCE AND HEDGING

σ一种+乙2=σ一种2+σ乙2+2ρ一种乙σ一种σ乙

σ一种+乙2=2σ2(1+ρ一种乙) 在哪里 σ一种2=σ乙2=σ2

ρ一种乙=0⇒σ一种+乙=2σ

σ是=nσ 当且当 ρ一世j=0∀一世≠j

μ=和[X]

μķ=和[(X−μ)ķ]

## 广义线性模型代考

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。