### 统计代写|工程统计作业代写Engineering Statistics代考|Discrete Distributions

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## 统计代写|工程统计作业代写Engineering Statistics代考|Discrete Distributions

There are two classes of distributions: Discrete and continuous. Discrete distributions are used to describe data that can have only discrete values. Such data have a specific probability associated with each value of the random variable. There are distinct and measurable step changes associated with each value of the variable. Some examples of discrete variables are the size of the last raise you received (it was not in fractions of a cent), the score of the last sporting event you watched, the number of personal protective equipment items available to you on your job, the number of first-quality computer chips on a silicon wafer, the number of defects in a skein of yarn, the energy of electrons in a particular quantum state, the number of raindrops that fall onto a square inch of land, etc.

The variable $x_{i}$ represents the count of events in the $i$ th category. The categories are mutually exclusive, such as alphabet letters, or pass/fail. The value of $x_{i}$ is an integer number. Looking at this paragraph, if $I=1$ represents the occurrence of the letter “a” and $I=2$ that of the letter ” $\mathrm{b}$ “, then the value of $x_{1}=18$ and $x_{2}=4$.

Probability density functions, $p d f\left(x_{i}\right)$ or simply $f\left(x_{i}\right)$, are associated with distributions of discrete variables, $x_{i}$ represent the probability of possible values of the ith data category. For example, if you flip a coin you expect $k=2$, two outcomes, Head and Tail, or 0 and 1 . If the first classification of $x_{1}=$ Head, then $f\left(x_{1}\right)=0.5$. All such probability functions have the following properties:

1. $x_{i}$ are the discrete possible values of a variable $X$, and $x_{\mathrm{i}}$ is the $i$ th of the $k$ finite values of the outcome. Usually, the index $i$ places the $x_{i}$ values in ascending order.
2. The probability functions are mathematical models of the population, of the infinity of possible samples, not of a finite sample of $k$ number of values.
3. $f\left(x_{i}\right)$ is the frequency, the probability of occurrence that a value $x_{i}$ will occur. It is positive and real for each $x_{\dot{r}} f\left(x_{i}\right)=\lim {n \rightarrow \infty}\left{n{i} / n\right}$.
4. $\sum_{i=1}^{k} f\left(x_{i}\right)=1$ where $k$ is the number of categories.
5. $P(E)=\sum f\left(x_{i}\right)$ where the sum includes all $x_{i}$ in the event $E$.
These definitions illustrate the notation we use throughout this book. We use capital Latin letters for populations and lowercase Latin letters for particular numerical observation values from the populations. Lowercase Greek letters are used for population parameters. Point and cumulative distributions are identified by $f$ and $F$ (or alternately CDF) respectively. $P$ stands for “probability of …”. We are using the conventional notation for discrete distributions: $x$ in the summations of the cumulative distribution functions of discrete distributions sometimes represents the number of items in a class (group, collection, etc.) or at other times, $x$ represents the numerical value that quantifies the class. By using this notation, the formulas in this book are consistent with those you may find in other statistics books. We state this as a warning, because in conventional notation for variables, $x$ means the value of the variable as opposed to the number of occurrences in a category. The cumulative distribution function (CDF) is a function $F\left(x_{n}\right)$ obtained from the probability function and is defined for the values of $x_{i}$ of the random variable $X$ by
$$F\left(x_{r}\right)=F\left(x_{r}\right)=P\left(X \leq x_{r}\right)= \begin{cases}0 & \text { for } X<x_{1} \ \sum_{i=1}^{r} f\left(x_{i}\right) & \text { where } x_{1} \leq X \leq x \ 1 & \text { for } X \geq x_{n}\end{cases}$$

## 统计代写|工程统计作业代写Engineering Statistics代考|Discrete Uniform Distribution

When each discrete event has the same likelihood (probability) of occurring, the probability function is given by
$$f\left(x_{i}\right)=\frac{1}{n}, \quad 1 \leq i<n$$
where $n$ is the number of discrete values for $x$. For the cumulative discrete distribution function,
$$F\left(x_{i}\right)=P\left(X \leq x_{i}\right)=\frac{i}{n}$$
where $x_{1}<x_{2}<x_{3} \ldots<x_{n}$
A classic example is that of rolling a cubical die. The $n=6$ categories of possible outcomes are equally probable.

The $X$ in Equation (3.6) may represent either a dimensionless counting number ( 7 bolts), a category ( 3 Heads) or a dimensional real number (last raise was $\$ 437.25 / \mathrm{month}$); however,$X$must be limited to a finite number,$n$, of discrete values. For the raise example, the discrete values are multiples of$1 \mathrm{c} / \mathrm{month}$. If the maximum possible raise could have been$\$600.00 /$ month, then $n=600.00 / .01+1=60,001$ (we cannot exclude the zero-raise event). Consequently, $x_{10,000}$ represents the 10,000 th value of $X$, which is $\$ 99.99 / \mathrm{mon}$th. Figure$3.1$illustrates the discrete uniform distribution for$n=5$, and the corresponding cumulative discrete uniform distribution, also for$n=5$. Recognizing that each$x_{i}$value has the same probability, or frequency of occurring,$f\left(x_{i}\right)=f\left(x_{j}\right)$, the mean and variance of the discrete uniform distribution are $$\begin{gathered} \mu=\frac{1}{n} \sum_{i=1}^{n} x_{i} \ \sigma^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2} \end{gathered}$$ If the$x_{i}$values are also equally incremented between$x_{1}=a$and$x_{n}=b$, so that$x_{j+1}-x_{j}=\Delta x=(b-a) / n$(such as with a die which has sides with values of$1,2,3 \ldots, 6$, where$a=1$and$b=6$) then ## 统计代写|工程统计作业代写Engineering Statistics代考|Binomial Distribution A discrete distribution called the binomial occurs when any observation can be placed in only one of two mutually exclusive categories, such as greater-than or less-than-or-equalto, safe or unsafe, hot or cold, on or off, 0 or 1, pass or fail, Heads or Tails, etc. Although these characteristics are qualitative, the distribution can be made quantitative by assigning the values 0 and 1 to the two categories. The method of assignment is immaterial so long as it is consistent. Customarily, the categories are labeled success (value$=1$) and failure (value$=0$). If$p=$probability of success and$q=1-p=$probability of failure in one trial of the experiment (one observation), the probability of exactly$x$number of successes in$n$trials can be described by the corresponding term of the binomial expansion, or $$f(x \mid n)=\left(\begin{array}{l} n \ x \end{array}\right) p^{x} q^{n-x} \equiv \frac{n !}{x !(n-x) !} p^{x} q^{n-x}, x=0,1,2, \ldots, n$$ where$X$may only have integer values. Note: The$\left(\begin{array}{l}n \ x\end{array}\right)$symbol does not mean$n$divided by$x$, it represents$\frac{n !}{x !(n-x) !}$, which is the number of combinations (ways) of having$x$occur in$n$trials. If$n=4$and$x=2$then$\left(\begin{array}{l}n \ x\end{array}\right)=\frac{4 !}{2 !(4-2) !}=\frac{4 \times 3 \times 2 \times 1}{2 \times 1(2 \times 1)}=6$. The six possible success-fail patterns could be 1100,1010 , 1001, 0110, 0101, and$0011 .$Note: The variable$x$represents the numerical count in a particular category, it is not the value of the category. Note: When$n$is large, the factorial terms become large, and direct calculation of either the numerator or denominator can result in digital overflow. Fortunately, the number of integers in the numerator and denominator is equal, there are$n$digits in each, and a best way to calculate the ratio is to alternate dividing and multiplying. But, many software packages provide convenient functions. In Excel the function is$f(x \mid n)=\operatorname{BINOMIAL} \cdot \mathrm{DIST}(x, n, p, 0)$. The binomial cumulative distribution function is $$F\left(x_{i} \mid n\right)=P\left(X \leq x_{i} \mid n\right)=\sum_{k=0}^{x_{i}}\left(\begin{array}{l} n \ k \end{array}\right) p^{k}(1-p)^{n-k}, i=0,1,2, \ldots, n$$ where$X$may have only integer values, for selected values of$n$and$p$. The notation (something$\mid n)$means “something given the value of$n^{\prime \prime}$. In Excel$F\left(x_{i} \mid n\right)=\operatorname{BINOMIAL}$.DIST$(x, n, p, 1)$. One can compute other probabilities such as$P\left(x_{i} \leq X \leq x_{i}\right)$, indicating the probability that an observation value,$X$, would be between and including$x_{i}$and$x_{\dot{r}}$$$P\left(x_{i} \leq X \leq x_{j}\right)=P\left(X \leq x_{j}\right)-P\left(X \leq x_{i-1}\right)=F\left(x_{j}\right)-F\left(x_{i-1}\right)$$ The best way to explain the use of Equation (3.11) is by use of a brief example. If you want$P(10 \leq X \leq 20)$, you need to exclude all values of$X$which are not in the probability specification. In this case, we want to include only the values$X=10,11,12, \ldots, 20$. The values of$X=0,1,2, \ldots, 9$must be excluded. As$x_{1}=10$, to exclude values below 10 , we must use$\left(x_{1}-1\right)=(10-1)=9$as the index. Specific values, such that the probability will be exactly$s$successes in$n$trials, can be found from $$P(X=s \mid n)=P(X \leq s \mid n)-P(X \leq(s-1) \mid n)$$ ## 工程统计代写 ## 统计代写|工程统计作业代写Engineering Statistics代考|Discrete Distributions 有两类分布：离散和连续。离散分布用于描述只能具有离散值的数据。这样的数据具有与随机变量的每个值相关联的特定概率。变量的每个值都存在明显且可测量的阶跃变化。离散变量的一些例子是你上一次加薪的大小（不是几分之一）、你最后一次观看体育赛事的分数、你在工作中可用的个人防护装备数量、硅晶片上第一质量的计算机芯片的数量，一束纱线中的缺陷数量，特定量子态的电子能量，落在一平方英寸土地上的雨滴数量等。 变量X一世表示事件的计数一世类别。类别是互斥的，例如字母或通过/失败。的价值X一世是一个整数。看这一段，如果一世=1表示字母“a”的出现和一世=2那封信”b“，那么值X1=18和X2=4. 概率密度函数，pdF(X一世)或者干脆F(X一世)，与离散变量的分布相关，X一世表示第 i 个数据类别的可能值的概率。例如，如果你掷硬币，你期望ķ=2，两个结果，头和尾，或 0 和 1。如果第一个分类X1=头，然后F(X1)=0.5. 所有这些概率函数都具有以下性质： 1. X一世是变量的离散可能值X， 和X一世是个一世的第ķ结果的有限值。通常，索引一世将X一世值按升序排列。 2. 概率函数是总体的数学模型，是无限可能样本的数学模型，而不是有限样本的数学模型。ķ值的数量。 3. F(X一世)是频率，一个值出现的概率X一世会发生。对于每个$x_{\dot{r}} f\left(x_{i}\right)=\lim {n \rightarrow \infty}\left{n {i} / n\right}$都是正实数. 4. ∑一世=1ķF(X一世)=1在哪里ķ是类别的数量。 5. 磷(和)=∑F(X一世)其中总和包括所有X一世在事件中和. 这些定义说明了我们在本书中使用的符号。我们使用大写拉丁字母表示总体，使用小写拉丁字母表示来自总体的特定数值观测值。小写希腊字母用于人口参数。点分布和累积分布由F和F（或交替CDF）分别。磷代表“……的概率”。我们对离散分布使用传统表示法：X在离散分布的累积分布函数的总和中有时表示一个类（组、集合等）或其他时候的项目数，X表示量化类的数值。通过使用这种表示法，本书中的公式与您在其他统计书籍中可能找到的公式是一致的。我们将此声明为警告，因为在变量的传统表示法中，X表示变量的值，而不是类别中的出现次数。累积分布函数 (CDF) 是一个函数F(Xn)从概率函数获得并定义为X一世随机变量X经过 F(Xr)=F(Xr)=磷(X≤Xr)={0 为了 X<X1 ∑一世=1rF(X一世) 在哪里 X1≤X≤X 1 为了 X≥Xn ## 统计代写|工程统计作业代写Engineering Statistics代考|Discrete Uniform Distribution 当每个离散事件具有相同的发生可能性（概率）时，概率函数由下式给出 F(X一世)=1n,1≤一世<n 在哪里n是离散值的数量X. 对于累积离散分布函数， F(X一世)=磷(X≤X一世)=一世n 在哪里X1<X2<X3…<Xn 一个典型的例子是掷骰子。这n=6可能结果的类别同样可能。 这X等式 (3.6) 中的数字可以表示无量纲计数（7 个螺栓）、类别（3 个 Heads）或有量纲实数（最后一次提升是$437.25/米这n吨H); 然而，X必须限制在一个有限的数量，n, 离散值。对于 raise 示例，离散值是1C/米这n吨H. 如果最大可能的加薪可能是$600.00/月，然后n=600.00/.01+1=60,001（我们不能排除零加注事件）。最后，X10,000代表第 10,000 个值X，即$99.99/米这nth。

μ=1n∑一世=1nX一世 σ2=1n∑一世=1n(X一世−μ)2

## 统计代写|工程统计作业代写Engineering Statistics代考|Binomial Distribution

F(X∣n)=(n X)pXqn−X≡n!X!(n−X)!pXqn−X,X=0,1,2,…,n

F(X一世∣n)=磷(X≤X一世∣n)=∑ķ=0X一世(n ķ)pķ(1−p)n−ķ,一世=0,1,2,…,n

## 有限元方法代写

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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。