经济代写|精算科学代写Actuarial Science代考|CRN27135

statistics-lab™ 为您的留学生涯保驾护航 在代写精算科学Actuarial Science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。我们的专家在代写精算科学Actuarial Science代写方面经验极为丰富，各种代写精算科学Actuarial Science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

经济代写|精算科学代写Actuarial Science代考|Approximations Using Taylor Series

If $f(x)$ is a function which has derivatives of all orders $\left(f^{\prime}, f^{\prime \prime}, f^{\prime \prime \prime}\right.$ etc., all exist) it can be shown that (under certain restrictions) $f(x)$ can be computed as an infinite sum of terms involving its derivatives.
$$f(x)=f(a)+f^{\prime}(a)(x-a)^{2}+\frac{f^{(2)}(a)}{2 !}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\cdots$$
In the expression above $f^{(n)}(a)$ refers to the $n^{\text {th }}$ derivative of $f$ evaluated at $a$. We can compute approximate values of $f(x)$ near a known value $f(a)$ by using the first few terms in 1.12.

Example 1.6: Use the first four terms of Equation $1.12$ and $a=0$ to approximate $\sin (x)$
Solution:
\begin{aligned} \sin (x) &=\sin (0)+\sin ^{\prime}(0)(x-0)+\frac{\sin ^{\prime \prime}(0)}{2 !}(x-0)^{2}+\cdots+\frac{\sin ^{\prime \prime \prime}(0)}{3 !}(x-0)^{3} \ &=0+\cos (0)(x-0)+\frac{-\sin (0)}{2 !}(x-0)^{2}+\frac{-\cos (0)}{3 !}(x-0)^{2} \ &=x-\frac{x^{3}}{6} \end{aligned}

经济代写|精算科学代写Actuarial Science代考|Measuring Interest

Perhaps the most common financial transaction is the investment of a certain amount of money at a specified rate of interest. A person might deposit money in a savings account with the expectation of earning interest on the amount deposited. Conversely, a bank makes loans with the expectation of being paid interest in addition to repayment of the principal. The term principal refers to the value of the loan/deposit at the time the transaction is made. The term present value is also used in this context.

In each case the interest is paid in compensation for the use of funds during the period of the transaction. The initial deposit or loan is called the principal and the total amount paid back after a period of time is called the accumulated value. The difference between the accumulated value and the initial deposit is the amount of interest.

Example 2.1 A loan of $\$ 1,000$is paid off with ten equal payments of$\$120$ each. What is the principal for this transaction? What is the amount of interest paid? Can we compute the interest rate for this transaction?

Solution: The principal is the amount of the loan: $\$ 1,000$. The payments total$\$1,200$ on a principal of $\$ 1,000$so the interest paid is$\$200$. Although $\$ 200$is$20 \%$of$\$1,000$, the interest rate on this loan is almost certainly not $20 \%$. To compute the interest rate we need to know when the payments took place. We will discuss the computation of interest rates for loans in Chapter $5 .$
We begin this chapter by discussing the various ways interest is computed when a single deposit is made and later withdrawn. Later, we will discuss those (much more common) cases where deposits and withdrawals occur throughout the period of an investment.

经济代写|精算科学代写Actuarial Science代考|Approximations Using Taylor Series

$$f(x)=f(a)+f^{\prime}(a)(x-a)^{2}+\frac{f^{(2)}(a)}{2 !}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\cdots$$

$$\sin (x)=\sin (0)+\sin ^{\prime}(0)(x-0)+\frac{\sin ^{\prime \prime}(0)}{2 !}(x-0)^{2}+\cdots+\frac{\sin ^{\prime \prime \prime}(0)}{3 !}(x-0)^{3} \quad=0+\cos (0)(x$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

经济代写|精算科学代写Actuarial Science代考|BSCHONS

statistics-lab™ 为您的留学生涯保驾护航 在代写精算科学Actuarial Science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。我们的专家在代写精算科学Actuarial Science代写方面经验极为丰富，各种代写精算科学Actuarial Science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

经济代写|精算科学代写Actuarial Science代考|Approximation Techniques

In many cases, we will need to solve equations for which no direct method applies. You are probably familiar with the quadratic formula: The solutions to $a x^{2}+b x+c=0$ are
$$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$$
There are similar equations for polynomials of degrees 3 and 4 , but no such formula exists for polynomials of degree 5 or higher. In some cases, we can reduce a higher degree polynomial to a quadratic, but these techniques won’t always work. As a result, we will utilize approximating techniques to solve such equations. We will use four methods.
a) Excel’s financial functions.
b) Newton’s Method (not used much anymore, provided as an historical note).
c) MAPLE (very powerful tool, but requires interpretation of results); MAPLE seems little used by financial folk.
d) TI Calculator internal calculation. Along with Excel, this will be the tool you will use most often in “the real world.”

经济代写|精算科学代写Actuarial Science代考|Newton’s Method

Isaac Newton (1643-1727), an English philosopher and mathematician, did important work in both physics and calculus. His method for approximating roots to polynomials is a very nice application of the tangent line. Joseph Raphson (1648-1715), also English, was made a member of the Royal Society prior to his graduation from Cambridge. See more about these two at the MacTutor History of Mathematics site: http://www-history.mes.standrews.ac.uk/index.html

Newton’s Method solves the equation $f(x)=0$ using an iteration technique. An iteration technique involves three stages:
1) Determining an initial guess (or approximation) called $x_{0}$,
2) Constructing an algorithm to compute $x_{i+1}$ in terms of $x_{i}$,
3) A proof that the sequence $x_{n}$ converges to the required value, in our case a solution of the equation $f(x)=0$.

The process starts with the initial approximation $x_{0}$ and then computes $x_{1}$, $x_{2}$, etc., until a desired degree of accuracy is attained. We will discuss how to make an educated guess (the $x_{0}$ ) in the context of specific problems ${ }^{4}$. At this point, we are interested only in describing how Newton’s Method generates the iteration sequence in 2). A proof that the method works is beyond the scope of this text – consult an Advanced Calculus text, if you would like to see a proof.

To create the sequence of approximations using the Newton-Raphson Method, we start with a reasonable first approximation, $x_{0}$. Often this is done by using a graphing calculator to graph the function and then reading off an estimate from the graph. To find $x_{1}$, we first construct the tangent line to the graph of $f$ at the point $\left(x_{0}, f\left(x_{0}\right)\right)$. The second estimate, $x_{1}$, is the $x$-intercept of this tangent line.

经济代写|精算科学代写Actuarial Science代考|Approximation Techniques

$$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$$

a) Excel 的财务功能。
b) 牛顿法 (不再使用太多，作为历史记录提供)。
c) MAPLE（非常强大的工具，但需要解释结果) ；MAPLE 似乎很少被金融界人士使用。
d) TI 计算器内部计算。与 Excel 一起，这将是您在“现实世界”中最常使用的工具。

经济代写|精算科学代写Actuarial Science代考|Newton’s Method

1) 确定初始猜测（或近似值），称为X0,
2) 构造一个算法来计算X一世+1按照X一世,
3) 证明该序列Xn收敛到所需的值，在我们的例子中是方程的解F(X)=0.

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

经济代写|精算科学代写Actuarial Science代考|Best20

statistics-lab™ 为您的留学生涯保驾护航 在代写精算科学Actuarial Science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。我们的专家在代写精算科学Actuarial Science代写方面经验极为丰富，各种代写精算科学Actuarial Science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

经济代写|精算科学代写Actuarial Science代考|Geometric Series

If we compute the sum of the values of the payments at the current time, the result is called the present value (PV) of the annuity. If we compute the accumulated values of the payments at some time in the future, the result is called the future value (FV) of the annuity. In either case, we will usually end up with a geometric series (the sum of a sequence where each term is a constant multiple of the preceding term) and so need the formula for the sum of such a series:
$$\sum_{i=0}^{n-1} a v^{i}=a+a v+a v^{2}+\cdots+a v^{n-1}=a \frac{1-v^{n}}{1-v}$$
Here $a$ is the initial term and $v$ is the common multiple ${ }^{1}$.
If $|v|<1$ then $\lim {n \rightarrow \infty} v^{n}=0$ and we can compute the sum of an infinite series of payments (called a perpetuity) as well: $$\sum{i=0}^{\infty} a v^{i}=\lim _{n \rightarrow \infty} a \frac{1-v^{n}}{1-v}=\frac{a}{1-v}$$

Using Equations $1.1$ and $1.2$ can be a bit tricky as not all series start at $i=0$. The most direct way to deal with this is to write down a few terms of the series you are dealing with and match them up with Equation $1.1$ or Equation 1.2. Note that you don’t need to figure out the last term since
\begin{aligned} &a=\text { first term } \ &v=\text { common multiple } \ &n=\text { number of terms. } \end{aligned}

经济代写|精算科学代写Actuarial Science代考|Arithmetic Series

An arithmetic series is created by adding the terms of a sequence where a constant (denoted by $d$ in the formula below) is added to each term to get the next term. In the case of an arithmetic series we have
$$a+(a+d)+(a+2 d)+\cdots+(a+(n-1) d)=\frac{n(2 a+(n-1) d)}{2}$$
Example 1.4: In the simplest case $a=d=1$ and we have the formula Carl Friederich Gauss supposedly proved at age six.
$$\sum_{i=1}^{n} i=1+2+3+\cdots+n=\frac{n(n+1)}{2}$$
In some cases, we will need to deal with a combination of an arithmetic and a geometric series:
$$A=P v+(P+Q) v^{2}+(P+2 Q) v^{3}+\cdots+(P+(n-1) Q) v^{n}$$
This situation (which we will refer to as a $P$ – $Q$ Series) arises when we have an annuity ${ }^{3}$ which starts with an initial payment which is then incremented by $Q$ at the end of each subsequent period ( $Q$ can be positive or negative). In many cases $Q$ is added to account for inflation. To simplify this expression we first divide both sides by $v$, obtaining:
$$\frac{A}{v}=P+(P+Q) v+(P+2 Q) v^{2}+(P+3 Q) v^{3}+\cdots+(P+(n-1) Q) v^{n-1}$$

经济代写|精算科学代写Actuarial Science代考|Geometric Series

$$\sum_{i=0}^{n-1} a v^{i}=a+a v+a v^{2}+\cdots+a v^{n-1}=a \frac{1-v^{n}}{1-v}$$

$$\sum i=0^{\infty} a v^{i}=\lim _{n \rightarrow \infty} a \frac{1-v^{n}}{1-v}=\frac{a}{1-v}$$

$$a=\text { first term } \quad v=\text { common multiple } n=\text { number of terms. }$$

经济代写|精算科学代写Actuarial Science代考|Arithmetic Series

$$a+(a+d)+(a+2 d)+\cdots+(a+(n-1) d)=\frac{n(2 a+(n-1) d)}{2}$$

$$\sum_{i=1}^{n} i=1+2+3+\cdots+n=\frac{n(n+1)}{2}$$

$$A=P v+(P+Q) v^{2}+(P+2 Q) v^{3}+\cdots+(P+(n-1) Q) v^{n}$$

$$\frac{A}{v}=P+(P+Q) v+(P+2 Q) v^{2}+(P+3 Q) v^{3}+\cdots+(P+(n-1) Q) v^{n-1}$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。