• 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

Many people have no difficulty performing the basic arithmetic operations (addition, subtraction, multiplication, and division) on positive numbers, but find similar operations on negative numbers mind-boggling. These operations typically are much easier to understand as they relate to profit and loss in commercial settings

In business, among many other uses, positive numbers represent profits, and negative numbers represent losses. In particular, $-\$ 5.00$denotes a loss of$\$5.00,-\$ 10.25$denotes a loss of$\$10.25$, and $+\$ 3.00$denotes a profit or gain of \$3.00. By convention, a number without a sign is considered positive. Therefore, $3=+3,5=+5$, and $9.75=+9.75$.

In the context of profits and losses, the addition sign is read “followed by.” Then, $-3+5$ is a $\$ 3$loss followed by a$\$5$ gain; the result is a net gain of $\$ 2$, so$-3+5=2$. For some, the process is clarified when viewed in the context of betting at a horse race. If a person loses$\$3$ on the first race and then wins $\$ 5$on the second race, he or she will have a net gain of$\$2$. Again, $-3+5=2$.
To calculate $3+(-7)$, we reason similarly. A $\$ 3$profit followed by a$\$7$ loss results in a net loss of $\$ 4$. Alternatively, if a person wins$\$3$ on the first race but loses $\$ 7$on the second race, he or she will then be behind by$\$4$. Either way, $3+(-7)=-4$.

The same reasoning is valid when adding two negative numbers. The quantity $(-10)+(-8)$ denotes a $\$ 10$loss followed by an$\$8$ loss. Or it can be viewed as a person losing $\$ 10$on the first bet and then losing$\$8$ on the second bet. The end result is the same, a total loss of $\$ 18$. Therefore$-10+(-8)=-18. Viewed as profits and losses, the following results should be straight forward: \begin{aligned} 5+7 &=12 \ -9+3 &=-6 \ 2+(-4) &=-2 \ 8+(-5) &=3 \ -7+(-8) &=-15 \end{aligned} The multiplication of signed numbers is a two-step operation. The first step is to multiply the numbers disregarding any negative signs (treat all numbers as positive). The second step is to determine the appropriate sign for the result. Here the following rules apply: ## 商科代写|商业数学代写business mathematics代考|EXPONENTS Exponents provide a convenient notation for representing the product of a number times itself many times. For example, consider the following, which are valid for any signed numbera(either positive or negative) \begin{aligned} &a^{2}=(a)(a) \ &a^{3}=(a)(a)(a) \ &a^{4}=(a)(a)(a)(a) \end{aligned} The definition of a number, denoted asa$, raised to the$n$th power, where$n$denotes a nonnegative integer (whole number), is given by $$a^{n}=(a)(a)(a)(a) \ldots \ldots(a) \leftarrow n \text { values of a multiplied together }$$ The quantity a” is typically read as either “the$n t h$power of$a, “$or ”$a$to the$n$th.” For example,$5^{2}=(5)(5)=25$is read as ” 5 squared is 5 times 5 equals 25 “$(-4)^{3}=(-4)(-4)(-4)=-64$is read as ”$-4$cubed is$-4$times$-4$times$-4$equals$-64$“$(-1 / 3)^{4}=(-1 / 3)(-1 / 3)(-1 / 3)(-1 / 3)=1 / 81$is read as$1 / 3$to the$4^{\text {th }}$power equals$\left.1 / 81^{\prime \prime}\right)$and$\begin{aligned} 2^{10}=(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)=& 1024 \text { is read as ” } 2 \text { to the } 10^{\text {th }} \text { power } \\left.\text { equals } 1024^{\prime \prime}\right) \end{aligned}$Notice that it is much easier to write$2^{10}$, than list 2 ten times. Additionally, as most calculators have an exponential function (usually with keys having either a$y^{x}$or${ }^{\wedge}$notation) it is easier to calculate the final numerical value using the designated exponential key than entering and multiplying the given number$n$times. One consequence of the exponential definition is the property $$\left(a^{n}\right)\left(a^{m}\right)=a^{n+m}$$ where$n$and$mare positive integers. For example, \begin{aligned} \left(6^{2}\right)\left(6^{3}\right) &=[(6)(6)][(6)(6)(6)]=(6)(6)(6)(6)(6)=6^{5}=6^{2+3} \ (-2)^{4}(-2)^{3} &=[(-2)(-2)(-2)(-2)][(-2)(-2)(-2)]=(-2)^{7}=(-2)^{4+3} \end{aligned} and $$\left(-\frac{1}{3}\right)^{4}\left(-\frac{1}{3}\right)^{2}=\left[\left(-\frac{1}{3}\right)\left(-\frac{1}{3}\right)\left(-\frac{1}{3}\right)\left(-\frac{1}{3}\right)\right]\left[\left(-\frac{1}{3}\right)\left(-\frac{1}{3}\right)\right]=\left(-\frac{1}{3}\right)^{6}=\left(-\frac{1}{3}\right)^{4+2}$$ Equation1.1$is valid only if the left side of the equation is a number raised to a power times that same number raised to a power. The formula is not valid if one$a$in Equation$1.1$is replaced by another number$b$. For example, Equation$1.1$is not applicable to the product$(2)^{5}(3)^{4}$. A second useful property of powers is $$\left(a^{n}\right)^{m}=a^{n m}$$ ## 商科代写|商业数学代写business mathematics代考|BASICS OF SOLVING EQUATIONS One major use of arithmetic operations on signed numbers and exponents is solving an equation for the unknown quantity. Any signed number that satisfies an equation (makes it true) is called a solution to the equation. For example, a value of$x$that satisfies the equation$-2 x=10$is a solution to the equation. Similarly, a value of$y$that satisfies the equation$2-y=4$is a solution to the equation. An equation that has a solution is called a conditional equation, while one that does not have a solution is called an inconsistent equation. An equation, such as$x=x$, for which any number is a solution is called an identity equation. In finding solutions (that is, one or more values that satisfy the equation), you should be aware of two notational conventions that are universally followed when writing equations with unknowns. First, parentheses are omitted for the product of a known number and an unknown quantity. For example,$(8)(y)$is written as$8 y$and$(-3)(x)$is written as$-3 x$. Secondly, if the product involves a 1 , the 1 is omitted and simply understood. Accordingly, both (1)y and$l y$are written as$y$, both (1)$x$and$l x$are written as$x$, both (1)p and$1 p$are written as$p$, and so on. The same convention holds for$-1$. Thus, for example, both$(-1) y$and$-l y$are written as$-y$, and both$(-1) x$and$-1 x$are written as$-x$. A numerical value for an unknown quantity in an equation is a solution for the equation if that value, when substituted for the unknown, makes the equality valid. For example, to determine whether$x=4$is a solution of$-2 x=10$, substitute$x=4$into the equation. Because$-2 x=-2(4)=-8$, which does not equal 10 , the value 4 is not a solution to the equation. ## 商业数学代考 ## 商科代写|商业数学代写business mathematics代考|SIGNED NUMBERS 许多人对正数执行基本的算术运算（加法、减法、乘法和除法）没有任何困难，但对负数的类似运算却令人难以置信。这些操作通常更容易理解，因为它们与商业环境中的损益有关 在商业中，在许多其他用途中，正数代表利润，负数代表损失。尤其是，−$5.00表示损失$5.00,−$10.25表示损失$10.25， 和+$3.00表示利润或收益3.00美元。按照惯例，没有符号的数字被认为是正数。所以，3=+3,5=+5， 和9.75=+9.75.

5+7=12 −9+3=−6 2+(−4)=−2 8+(−5)=3 −7+(−8)=−15

52=(5)(5)=25读作“5 的平方是 5 乘以 5 等于 25”
(−4)3=(−4)(−4)(−4)=−64读作“−4立方是−4次
−4次−4等于−64 “
(−1/3)4=(−1/3)(−1/3)(−1/3)(−1/3)=1/81读作1/3到4th 功率等于1/81′′)

\begin{aligned} 2^{10}=(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)=& 1024 \text { 读作” } 2 \text { 到 } 10^{\text {th }} \text { power } \\left.\text { equals } 1024^{\prime \prime}\right) \end{aligned}\begin{aligned} 2^{10}=(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)=& 1024 \text { 读作” } 2 \text { 到 } 10^{\text {th }} \text { power } \\left.\text { equals } 1024^{\prime \prime}\right) \end{aligned}

(一个n)(一个米)=一个n+米

(62)(63)=[(6)(6)][(6)(6)(6)]=(6)(6)(6)(6)(6)=65=62+3 (−2)4(−2)3=[(−2)(−2)(−2)(−2)][(−2)(−2)(−2)]=(−2)7=(−2)4+3

(−13)4(−13)2=[(−13)(−13)(−13)(−13)][(−13)(−13)]=(−13)6=(−13)4+2

(一个n)米=一个n米

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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。