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

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|COMBINATORICS

In elementary combinatorics, one typically learns about combinations and permutations. Combinations tell us how many ways we can arrange a number of objects, regardless of the order, whereas permutations tell us how many ways we can arrange a number of objects, taking into account the order.

As an example, assume we have three hedge funds, denoted X, Y, and
$Z$. We want to invest in two of the funds. How many different ways can we invest? We can invest in X and $Y, X$ and Z, or $Y$ and $Z$. That’s it.

In general, if we have $n$ objects and we want to choose $k$ of those objects, the number of combinations, $C(n, k)$, can be expressed as:
$$C(n, k)=\left(\begin{array}{l} n \ k \end{array}\right)=\frac{n !}{k !(n-k) !}$$
where $n !$ is $n$ factorial, such that:
$$n !=\left{\begin{array}{cc} 1 & n=0 \ n(n-1)(n-2) \cdots 1 & n>0 \end{array}\right.$$
In our example with the three hedge funds, we would substitute $n=3$ and $k=2$, to get three possible combinations.

What if the order mattered? What if instead of just choosing two funds, we needed to choose a first-place fund and a second-place fund? How many

ways could we do that? The answer is the number of permutations, which we express as:
$$P(n, k)=\frac{n !}{(n-k) !}$$
For each combination, there are $k$ ! ways in which the elements of that combination can be arranged. In our example, each time we choose two funds, there are two ways that we can order them, so we would expect twice as many permutations. This is indeed the case. Substituting $n=3$ and $k=2$ into Equation 1.18, we get six permutations, which is twice the number of combinations computed previously.

Combinations arise in a number of risk management applications. The binomial distribution, which we will introduce in Chapter 4 , is defined using combinations. The binomial distribution, in turn, can be used to model defaults in simple bond portfolios or to back-test Value at Risk (VaR) models, as we will see in Chapter $5 .$

Combinations are also central to the binomial theorem. Given two variables, $x$ and $y$, and a positive integer, $n$, the binomial theorem states:
$$(x+y)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \ k \end{array}\right) x^{n-k} y^{k}$$
For example:
$$(x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}$$
The binomial theorem can be useful when computing statistics such as variance, skewness, and kurtosis, which will be discussed in Chapter 3 .

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

Most people have a preference for present income over future income. They would rather have a dollar today than a dollar one year from now. This is why banks charge interest on loans, and why investors expect positive returns on their investments. Even in the absence of inflation, a rational person should prefer a dollar today to a dollar tomorrow. Looked at another way, we should require more than one dollar in the future to replace one dollar today.

In finance we often talk of discounting cash flows or future values. If we are discounting at a fixed rate, $R$, then the present value and future value are related as follows:
$$V_{t}=\frac{V_{t+n}}{(1+R)^{n}}$$
where $V_{t}$ is the value of the asset at time $t$ and $V_{t+n}$ is the value of the asset at time $t+n$. Because $R$ is positive, $V_{t}$ will necessarily be less than $V_{t+n}$. All else being equal, a higher discount rate will lead to a lower present value. Similarly, if the cash flow is further in the future-that is, $n$ is greater-then the present value will also be lower.

Rather than work with the discount rate, $R$, it is sometimes easier to work with a discount factor. In order to obtain the present value, we simply multiply the future value by the discount factor:
$$V_{t}=\left(\frac{1}{1+R}\right)^{n} V_{t+n}=\delta^{n} V_{t+n}$$
Because $\delta$ is less than one, $V_{t}$ will necessarily be less than $V_{t+n}$. Different authors refer to $\delta$ or $\delta^{n}$ as the discount factor. The concept is the same, and which convention to use should be clear from the context.

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

The ancient Greek philosopher Zeno, in one of his famous paradoxes, tried to prove that motion was an illusion. He reasoned that, in order to get anywhere, you first had to travel half the distance to your ultimate destination. Once you made it to the halfway point, though, you would still have to travel half the remaining distance. No matter how many of these half journeys you completed, there would always be another half journey left. You could never possibly reach your destination.

While Zeno’s reasoning turned out to be wrong, he was wrong in a very profound way. The infinitely decreasing distances that Zeno struggled with

foreshadowed calculus, with its concept of change on an infinitesimal scale. Also, an infinite series of a variety of types turn up in any number of fields. In finance, we are often faced with series that can be treated as infinite. Even when the series is long, but clearly finite, the same basic tools that we develop to handle infinite series can be deployed.

In the case of the original paradox, we are basically trying to calculate the following summation:
$$S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$$
What is $S$ equal to? If we tried the brute force approach, adding up all the terms, we would literally be working on the problem forever. Luckily, there is an easier way. The trick is to notice that multiplying both sides of the equation by $1 / 2$ has the exact same effect as subtracting $1 / 2$ from both sides.

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

C(n,ķ)=(n ķ)=n!ķ!(n−ķ)!

$$n !=\left{1n=0 n(n−1)(n−2)⋯1n>0\对。$$

(X+是)n=∑ķ=0n(n ķ)Xn−ķ是ķ

(X+是)3=X3+3X2是+3X是2+是3

## 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。