物理代写|统计力学代写Statistical mechanics代考|PHYS3034

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• 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

物理代写|统计力学代写Statistical mechanics代考|EVENTS, SAMPLE SPACE, AND PROBABILITY

The sample space of two coins tossed is $\Omega={H H, H T, T H, T T}$. One way to represent these outcomes would be to assign $H$ the number 1 and $T$ the number 0 , so that they’re given by the points $(1,1),(1,0),(0,1),(0,0)$ in the $x y$ plane; see Fig. 3.1. One doesn’t have to depict the sample space as in Fig. 3.1-one could mark off any four points on the $x$-axis for example. For three coins tosised there are eight outcomes; one could depict the sample space using a three-dimensional Cartesian space, or, again, mark off any eight points on the $x$-axis. Sample space is a useful mathematical concept for discussing probability. How we display these sets is a matter of convenience. It’s often simpler to display the sample space in an abstract manner. The right part of Fig. $3.1$ shows the 36 elements of $\Omega$ for the roll of two dice simply as points in a box.

Experiments that produce a finite number of outcomes, such as the roll of a die, have discrete sample spaces where the events can be represented as isolated points, as in Fig. 3.1. Probabilities defined on discrete sample spaces are referred to as discrete probabilities. Not every sample space is discrete. Continuous sample spaces are associated with experiments that produce a continuous range of possibilities, such as the heights of individuals in a certain population. Probabilities defined on continuous sample spaces are referred to as probability densities.

The individual elements of $\Omega$ are elementary events. ${ }^{1}$ The word event (not elementary event) is reserved for subsets of $\Omega$, aggregates of sample points. A subset $A$ of $\Omega$ is a set such that every element of $A$ is an element of $\Omega$, a relationship indicated $A \subset \Omega$. In tossing two coins, the event $A$ might be the occurrence of $T T$ or $H H ; A \subset \Omega$ is then the set of elementary events $A={T T, H H}$, where $\Omega={T T, H H, T H, H T}$. The terms “sample point” and “event” have an intuitive appeal, that, once specified for a given experiment, can be treated using the mathematics of point sets.

Consider events $A$ and $B$ (such as in Fig. 3.3), which have $N_{A}$ and $N_{B}$ sample points (elementary events). In $A \cup \perp$ there are $N_{A \cup B}=N_{A}+N_{B}-N_{A \cap B}$ elements, where $N_{A \cap B}$ is the number of elements of the intersection $A \cap B$, which must be subtracted to prevent overcounting. ${ }^{6}$ We then have using Eq. (3.1) the analogous formula for probabilities,
$$P(A \cup B)=P(A)+P(B)-P(A \cap B) .$$
If $A$ and $B$ have no sample points in common (mutually exclusive), $A \cap B=\emptyset$. In that case,
$$P(A \cup B)=P(A)+P(B) . \quad(A, B \text { mutually exclusive })$$
Equation (3.3) is used frequently in applications-it tells us that the probability of $A$ or $B$ is the sum of the probabilities when $A, B$ are mutually exclusive. It pays to get in the habit of noticing how many calculations stem from questions of the form “what is the probability of the occurrence of this or that or that?” There’s often an implicit “or” statement underlying calculations in physics. Equation (3.3) easily generalizes to more than two mutually exclusive events.

物理代写|统计力学代写Statistical mechanics代考|EVENTS, SAMPLE SPACE, AND PROBABILITY

$N_{A \cup B}=N_{A}+N_{B}-N_{A \cap B}$ 元素，其中 $N_{A \cap B}$ 是交点的元素个数 $A \cap B$, 必须减去以防止多算。 ${ }^{6}$ 然后我们使用 方程式。(3.1) 概率的类似公式，
$$P(A \cup B)=P(A)+P(B)-P(A \cap B) .$$

$$P(A \cup B)=P(A)+P(B) . \quad(A, B \text { mutually exclusive })$$

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。