概率论是研究随机性或不确定性等现象的 数学

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数据科学概率导论

这本书是大学概率论的入门教材。它有一个使命: 阐明我们在科学和工程中使用的概率工具的动机、直觉和含义。从超过五年的课程教学中,我提炼出了我认为是概率方法的核心。我把这本书放在数据科学的背景下,以强调数据(计算)和概率(理论)在我们这个时代的不可分离性。

地址: https://probability4datascience.com/index.html

概率论是电子工程和计算机科学中最有趣的学科之一。它将我们喜爱的工程原理与现实联系起来,这是一个充满不确定性的世界。然而,因为概率是一门非常成熟的学科,单是本科生的课本就可能在图书馆的书架上摆满好几排书。当文学如此丰富时,挑战就变成了一个人如何在深入细节的同时洞察到洞察力。例如,你们中的许多人以前使用过正态随机变量,但你们是否想过“钟形”是从哪里来的?每一门概率课都会教你抛硬币,但是“抛硬币”在今天的机器学习中有什么用呢?数据科学家使用泊松随机变量来模拟互联网流量,但是这个漂亮的泊松方程是从哪里来的呢?这本书的目的是填补这些知识的差距,这是所有数据科学学生必不可少的。

这就引出了本书的三个目标。(i) 动机: 在数学定义、定理、方程的海洋中,为什么我们要把时间花在这个主题上,而不是其他的? (ii) 直觉: 当进行推导时,在这些方程之外是否有几何解释或物理学?(iii) 言外之意: 当我们学习了一个话题后,我们可以解决哪些新问题?本书的目标读者是电子工程和计算机科学专业的本科生三、四年级和一年级研究生。先决条件是标准的本科线性代数和微积分,除了需要傅里叶变换的特征函数部分。一门信号与系统的本科课程就足够了,即使是在学习这本书的同时选修。

这本书的篇幅适合两学期的课程。教师被鼓励使用最适合他们的课程的章节集。例如,基本概率课程可以使用第1-5章作为主干。关于样本统计的第6章适合希望获得概率收敛理论见解的学生。关于回归的第七章和关于估计的第八章最适合学习机器学习和信号处理的学生。第9章讨论了对现代数据分析至关重要的置信区间和假设检验。第10章介绍了随机过程。我的随机过程方法更适合于信息处理和通信系统,这通常与电气工程专业的学生更相关。

本书特色:

涵盖范围广,从经典的概率论到现代数据分析技术 概念的几何和图形解释 与MATLAB / Python紧密集成 机器学习的实际应用

目录内容

Chapter 1 Mathematical Background Chapter 2 Probability Chapter 3 Discrete Random Variables Chapter 4 Continuous Random Variables Chapter 5 Joint Distributions Chapter 6 Sample Statistics Chapter 7 Regression Chapter 8 Estimation Chapter 9 Confidence and Hypothesis Chapter 10 Random Processes

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Whenever students use any drilling system the question arises how much of their learning is meaningful learning vs memorisation through repetition or rote learning. Although both types of learning have their place in an educational system it is important to be able to distinguish between these two approaches to learning and identify options which can dislodge students from rote learning and motivate them towards meaningful learning. The tutor-web is an online drilling system. The design aim of the system is learning rather than evaluation. This is done by presenting students with multiple-choice questions which are selected randomly but linked to the students' performance. The questions themselves can be generated for a specific topic by drawing correct and incorrect answers from a collection associated with a general problem statement or heading. With this generating process students may see the same question heading twice but be presented with all new answer options or a mixture of new and old answer options. Data from a course on probability theory and statistics, taught during COVID-19, are analysed to separate rote learning from meaningful learning. The analyses show non-rote learning, but even with large question databases, students' performance is better when they are presented with an answer option they have seen before. An element of rote learning is thus exhibited but a deeper learning is also demonstrated. The item database has been seeded with hints such that some questions contain clues to cue the students towards the correct answer. This ties in with the issue of meaningful learning versus rote learning since the hope is that a new hint will work as a cue to coax the student to think harder about the question rather than continue to employ rote learning. Preliminary results indicate that hints are particularly useful for students with poor performance metrics.

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