商科代写|商业数学代写business mathematics代考|SIGMA NOTATION

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我们提供的商业数学business mathematics及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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
商科代写|商业数学代写business mathematics代考|SIGMA NOTATION

商科代写|商业数学代写business mathematics代考|SIGMA NOTATION

In various parts of this text, we will need the sum of large numbers of terms. Sometimes we will need the actual total, and then we will have to sum the numbers physically. Other times, however, we will only need to indicate the appropriate sum. An example is a statement, “Yearly expenditures are the sum of weekly expenditures.” Here we are not explicitly calculating the total yearly expenditure but simply indicating that it is the sum of 52 numbers. In cases like this, there is a useful mathematical notation for indicating the appropriate sum.

Consider the case of a teacher who has a list of seven grades and wants their sum. If we denote the first grade as $G_{1}$, the second grade as $G_{2}$, and so on through the seventh grade, which we denote as $G_{7}$, the final sum can be given as
\text { Sum }=G_{1}+G_{2}+G_{3}+G_{4}+G_{5}+G_{6}+G_{7}
Of course, if we had 52 items to add as opposed to only 7 , writing an expression similar to Equation $1.7$ would be tedious indeed. A more convenient way to represent the right side of Equation $1.7$ symbolically as
\sum_{i=1}^{7} G_{i}
The capital Greek letter sigma $(\Sigma)$ denotes a sum. The term(s) after the sigma, in this case, the letter $G$, tells us what values are to be added, in this case, we are adding $G$ terms.

The quantity $i=1$ at the bottom of the sigma indicates where the sum is to start from; in this case, the sum starts with $G_{1}$, that is, $G_{i}$, with $i$ replaced with the starting value of 1 . The number at the top of the sigma indicates where the sum is to stop, which in this case is $G_{7}$. Intermediate values in the sum are obtained by replacing the subscript $i$ on individual $G$ terms with consecutive integers between the starting and ending numbers at the bottom and top of the sigma, respectively.
For example, the notation
\sum_{i=3}^{8} S_{i}
indicates a sum of $S$ terms. The sum starts with $S_{3}$, because the starting $i$ value is given as 3 at the bottom of the sigma sign, and ends at $S_{5}$, because the ending value of $i$ is given as 8 above the sigma sign. Thus, the notation
\sum_{i=3}^{s} S_{i}
is shorthand for

商科代写|商业数学代写business mathematics代考|NUMERICAL CONSIDERATIONS

Expressing numbers in decimal form is necessary for most commercial transactions. One reason involves money. All financial figures are given in decimal form; the numbers to the left of the decimal point represent dollars, and the numbers to the right of the decimal point represent cents. Quoting the cost of an item as $\$ 1.20$ is clearer than quoting six-fifths of a dollar. A second reason for using decimals is mathematical. It is easier to add $0.2$ and $1.5$ than it is to $1 / 5$ and $11 / 2$. In this section, we present two considerations when dealing with numerical values; rounding when dealing with dollar and cents commercial transactions and exponential notation that you will sometimes encounter when using a calculator.

The most common form of rounding is referred to as arithmetic rounding or just rounding for short. Here one first decides how many digits are to be retained and then looks at the next digit to the right. If it is greater than or equal to 5 , the previous digit (which is the last one being kept) is increased by 1 ; otherwise, no change is made. All digits to the right are then either set to zero or dropped.

As an example, consider the irrational number $\pi=3.1415926 \ldots$ To round this number to three decimal places, first look at the digit four places to the right of the decimal point. It is 5 , so we increase the previous digit by 1 and write $\pi=3.142$ rounded to three decimal places. To round the same number to two decimal places, first look at the digit three places to the right of the decimal point. It is 1 , which is less than 5 . Thus, $\pi=3.14$ rounded to two decimal places.

Rounding is equally applicable to finite decimals. Anytime one uses the above procedure to reduce the number of digits in a number, one is rounding. For example, $81.314=81.31$ rounded to two decimal places, $8595.72=8596$ rounded to units, and $0.0051624=0.0052$ rounded to four decimal places.
The second form of rounding is called rounding up. Here the last digit being kept in a number is automatically increased by 1 if any one of the discarded digits is not 0 . For example, $8.1403=8.15$ rounded up to two decimal places (note that one of the discarded digits is a 3 ; that is, not all of the discarded digits are zero), $1.38112=1.382$ rounded up to three decimal places, and $1.900=1.9$ rounded to one decimal place. In the last example, we did not increase the 9 because all the discarded digits were 0 .

商科代写|商业数学代写business mathematics代考|THE CARTESIAN COORDINATE SYSTEM

A graph is a diagram that represents data in an organized manner. Bar graphs, column graphs, line graphs, and pie charts, as shown in Figures $2.1$ through $2.3$, respectively, illustrate these types of graphs. ${ }^{1}$

In this chapter, we will focus our attention on line graphs, although, as shown in Section 2.6, this type of graph is easily converted into any of the other graph types shown in Figures $2.2$ and 2.3.

The most commonly used line graph is constructed as points on a twodimensional plane constructed on a Cartesian coordinate system. To understand this system, first, consider the map illustrated in Figure $2.4$ with streets running in either a north-south direction or an east-west direction.

By using the center of Broad and Market Streets as a reference (perhaps a motorist has stopped there for directions), it is easy to locate any other point on the map. The intersection of Elm Lane and Maple Street is two blocks west and one block south of the reference point. The light at Freeman Street and Valley Road is three blocks east and two blocks north of the reference point.

In each case, the new point on the map is uniquely determined from the reference point by two numbers and their direction.

The Cartesian coordinate system, which is also known as the Rectangular coordinate system, is a generalized version of the previous map. To construct this system, two intersecting perpendicular lines (forming an angle of 90 degrees with each other) are first drawn, as illustrated in Figure $2.5$. The horizontal line is often called the $x$-coordinate axis (or just the $x$-axis for short), while the vertical line is often called the $y$-coordinate axis (or $y$-axis for short). The intersection of these two axes is the origin, and it represents the reference point of the system.

商科代写|商业数学代写business mathematics代考|SIGMA NOTATION


商科代写|商业数学代写business mathematics代考|SIGMA NOTATION

在本书的各个部分,我们将需要大量术语的总和。有时我们需要实际的总数,然后我们必须对这些数字进行物理求和。然而,其他时候,我们只需要指出适当的总和。例如,“年度支出是每周支出的总和”。在这里,我们没有明确计算每年的总支出,而只是表明它是 52 个数字的总和。在这种情况下,有一个有用的数学符号来表示适当的总和。

考虑一个老师的例子,他有一个七个年级的列表并想要他们的总和。如果我们将一年级表示为G1, 二年级为G2,以此类推,一直到七年级,我们将其表示为G7,最终的总和可以给出为

 和 =G1+G2+G3+G4+G5+G6+G7
当然,如果我们要添加 52 项而不是只有 7 项,请编写类似于 Equation 的表达式1.7确实会很乏味。一种更方便的表示方程式右侧的方法1.7象征性地作为

大写希腊字母 sigma(Σ)表示总和。sigma 之后的术语,在这种情况下,字母G, 告诉我们要添加哪些值,在这种情况下,我们正在添加G条款。

数量一世=1sigma 的底部表示总和的起点;在这种情况下,总和以G1, 那是,G一世, 和一世替换为 1 的起始值。sigma 顶部的数字表示总和的停止位置,在本例中为G7. 和中的中间值是通过替换下标得到的一世在个人G分别在 sigma 的底部和顶部的开始和结束数字之间具有连续整数的项。

表示总和小号条款。总和开始于小号3,因为开始一世值在 sigma 符号的底部给出为 3,并在小号5, 因为结束值一世在 sigma 符号上方给出 8。因此,符号



商科代写|商业数学代写business mathematics代考|NUMERICAL CONSIDERATIONS

大多数商业交易都需要以十进制形式表示数字。一个原因与金钱有关。所有财务数字均以十进制形式给出;小数点左边的数字代表美元,小数点右边的数字代表美分。将项目的成本引用为$1.20比引用一美元的五分之六更清楚。使用小数的第二个原因是数学。更容易添加0.2和1.5而不是1/5和11/2. 在本节中,我们在处理数值时提出两个考虑因素;处理美元和美分商业交易时的四舍五入以及使用计算器时有时会遇到的指数符号。

最常见的舍入形式称为算术舍入或简称舍入。这里首先决定要保留多少位,然后查看右边的下一位。如果大于等于 5 ,则前一位(保留的最后一位)加 1 ;否则,不进行任何更改。然后,右边的所有数字要么设置为零,要么被丢弃。

例如,考虑无理数圆周率=3.1415926…要将这个数字四舍五入到小数点后三位,首先查看小数点右侧的四位数字。它是 5 ,所以我们将前一个数字加 1 并写圆周率=3.142四舍五入到小数点后三位。要将相同的数字四舍五入到小数点后两位,首先查看小数点右侧三位的数字。它是 1 ,小于 5 。因此,圆周率=3.14四舍五入到小数点后两位。

第二种形式的舍入称为向上舍入。如果任何一个丢弃的数字不是 0 ,则保留在数字中的最后一个数字会自动增加 1 。例如,8.1403=8.15四舍五入到小数点后两位(请注意,丢弃的数字之一是 3 ;也就是说,并非所有丢弃的数字都为零),1.38112=1.382四舍五入到小数点后三位,并且1.900=1.9四舍五入到小数点后一位。在最后一个例子中,我们没有增加 9 因为所有丢弃的数字都是 0 。

商科代写|商业数学代写business mathematics代考|THE CARTESIAN COORDINATE SYSTEM


在本章中,我们将把注意力集中在折线图上,尽管如第 2.6 节所示,这种类型的图很容易转换为图中所示的任何其他图类型2.2和 2.3。


通过使用 Broad Street 和 Market Streets 的中心作为参考(也许驾车者已经在那里停下来寻找方向),很容易找到地图上的任何其他点。Elm Lane 和 Maple Street 的交叉口位于参考点以西两个街区和以南一个街区。Freeman Street 和 Valley Road 的灯在参考点以东三个街区和以北两个街区。


笛卡尔坐标系,也称为直角坐标系,是以前地图的广义版本。为了构建这个系统,首先绘制两条相交的垂直线(彼此成 90 度角),如图所示2.5. 水平线通常称为X- 坐标轴(或只是X轴),而垂直线通常称为是- 坐标轴(或是简称-轴)。这两个轴的交点就是原点,它代表系统的参考点。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。