数学生对测度的执念
英文版: Claudio Landim at IMPA, Brazil.
https://www.bilibili.com/video/BV1EW411K7dN
中文版:antiRomance 解释陈世宏老师的测度论与概率论基础
https://www.bilibili.com/video/BV1pw411Z7Ds
张量的定义与张量的表示-2021年
https://www.bilibili.com/video/BV1Wp4y1H7KG
【张量分析】
https://www.bilibili.com/video/BV1SY4y1z7km
having a solid grasp of Measure Theory provides a more rigorous and comprehensive foundation
【使得现代概率论有了严谨,全面的基础】
可以对集合set 定义size大小,或volume容积。延申了长度,面积,容积的定义。length, area, volume
- 针对连续随机变量,测度论严谨定义了pdf 概率密度方程,这个pdf是这个连续随机变量(continuous r.v.)的分布陈述。
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大数定理,中心极限定理,概率论的核心。$\[X_i \stackrel{\text{i.i.d.}}{\sim} N(\mu, \sigma^2)\]$
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随机过程: 测度让Brownian motion布朗运动 和 Markov chains 连续随机游走和马尔科夫链成为了可能。
- 条件概率:一样
Measure spaces测度空间:
一个集合,sigma-代数,测度。
测度空间即是概率空间的基础,概率空间是一种特殊的测度空间。
Lebesgue measure and integration L测度和导数
– 这里没细看,之后写
Convergence theorem 收敛定理
Monotone Convergence Theorem, Fatou’s Lemma, and the Dominated Convergence Theorem.
==>Strong Law of Large Numbers and the Lebesgue Differentiation Theorem. 强大数公理,L求导定理
Radon-Nikodym theorem:
描述了两个测度之间的绝对连续性
absolute continuity
==>条件概率,条件期望 conditional probability and conditional expectation,
Ergodic theory 随时间变化,动态系统
CLT: 可以基于CLT, 高斯分布对未知数据分布进行估量,哪怕是未知分布。
even when the underlying distribution is unknown or not normally distributed.
Continuous random variables: Measure Theory allows for the proper definition of probability density functions, which are used to describe the probabilities associated with continuous random variables.
Limit theorems: Measure Theory provides the tools for understanding and proving key results like the Law of Large Numbers and the Central Limit Theorem, which are essential in Statistics.
Stochastic processes: Measure Theory helps in the formulation of stochastic processes, such as Brownian motion or Markov chains, which are widely used in various fields like finance, physics, and engineering.
Conditional probabilities and expectations: Measure Theory plays a crucial role in developing the concepts of conditional probability and conditional expectation, which are essential in many statistical applications.