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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Generating Values from an Arbitrary Distribution

The cumulative density function $F(X)$, where $X$ is either discrete or continuous, can be used to generate values drawn from the underlying discrete or continuous distribution $p\left(X_{d}\right)$ or $f\left(X_{c}\right)$, as illustrated in Figure 1.2. Consider a discrete random variable $X_{d}$ that takes on values $x_{1}, x_{2}, \ldots, x_{n}$ with probabilities $p\left(x_{i}\right)$. By definition, $F\left(x_{k}\right)=F\left(x_{k-1}\right)+p\left(x_{k}\right)$. Moreover, $F\left(X_{d}\right)$ always lies in the range $[0,1]$. Therefore, if we were to generate a random number $u$ with uniform probability in the range $[0,1]$, the probability that $u$ lies in the range $\left[F\left(x_{k-1}\right), F\left(x_{k}\right)\right]$ is $p\left(x_{k}\right)$. Moreover, $x_{k}=F^{-1}(u)$. Therefore, the procedure to generate values from the discrete distribution $p\left(X_{d}\right)$ is as follows: First, generate a random variable $u$ uniformly in the range $[0,1]$; second, compute $x_{k}=F^{-1}(u)$.

We can use a similar approach to generate values from a continuous random variable $X_{c}$ with associated density function $f\left(X_{c}\right)$. By definition, $F(x+\delta)=F(x)+$ $f(x) \delta$ for very small values of $\delta$. Moreover, $F\left(X_{c}\right)$ always lies in the range $[0,1]$. Therefore, if we were to generate a random number $u$ with uniform probability in the range $[0,1]$, the probability that $u$ lies in the range $[F(x), F(x+\delta)]$ is $f(x) \delta$, which means that $x=F^{-1}(u)$ is distributed according to the desired density function $f\left(X_{c}\right.$ ). Therefore, the procedure to generate values from the continuous distribution $f\left(X_{c}\right)$ is as follows: First, generate a random variable $u$ uniformly in the range $[0,1]$; second, compute $x=F^{-1}(u) .$

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Moments and Moment Generating Functions

Thus far, we have focused on elementary concepts of probability. To get to the next level of understanding, it is necessary to dive into the somewhat complex topic of moment generating functions. The moments of a distribution generalize its mean and variance. In this section, we will see how we can use a moment generating function (MGF) to compactly represent all the moments of a distribution. The moment generating function is interesting not only because it allows us to prove some useful results, such as the central limit theorem but also because it is similar in form to the Fourier and Laplace transforms, discussed in Chapter $5 .$

The moments of a distribution are a set of parameters that summarize it. Given a random variable $X$, its first moment about the origin, denoted $M_{0}^{1}$, is defined to be $E[X]$. Its second moment about the origin, denoted $M_{0}^{2}$, is defined as the expected value of the random variable $X^{2}$, or $E\left[X^{2}\right]$. In general, the $r$ th moment of $X$ about the origin, denoted $M_{0}^{r}$, is defined as $M_{0}^{r}=E\left[X^{r}\right]$.

We can similarly define the $r$ th moment about the mean, denoted $M_{\mu}^{r}$, by $E[(X-$ $\left.\mu)^{r}\right]$. Note that the variance of the distribution, denoted by $\sigma^{2}$, or $V[X]$, is the same as $M_{\mu}^{2}$. The third moment about the mean, $M_{\mu}^{3}$, is used to construct a measure of skewness, which describes whether the probability mass is more to the left or the right of the mean, compared to a normal distribution. The fourth moment about the mean, $M_{\mu}^{4}$, is used to construct a measure of peakedness, or kurtosis, which measures the “width” of a distribution.

The two definitions of a moment are related. For example, we have already seen that the variance of $X$, denoted $V[X]$, can be computed as $V[X]=E\left[X^{2}\right]-(E[X])^{2}$. Therefore, $M_{\mu}^{2}=M_{0}^{2} \quad\left(M_{0}^{1}\right)^{2}$. Similar relationships can be found between the higher moments by writing out the terms of the binomial expansion of $(X-\mu)^{r}$.

计算数学基础代考

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。

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