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| =Probability Space= | | {{Infobox |
| The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one of the greatest mathematician of the 20th century, who advanced various very different fields of mathematics.
| | |name = Infobox |
| | |bodystyle = |
| | |title = 大气环流 <br> |
| | General Circulation of the Atmosphere |
| | |titlestyle = |
|
| |
|
| {{Theorem|Definition (Probability Space)|
| | |headerstyle = background:#ccf; |
| A '''probability space''' is a triple <math>(\Omega,\Sigma,\Pr)</math>.
| | |labelstyle = background:#ddf; |
| *<math>\Omega</math> is a set, called the '''sample space'''.
| | |datastyle = |
| *<math>\Sigma\subseteq 2^{\Omega}</math> is the set of all '''events''', satisfying:
| |
| *:(K1). <math>\Omega\in\Sigma</math> and <math>\emptyset\in\Sigma</math>. (Existence of the ''certain'' event and the ''impossible'' event)
| |
| *:(K2). If <math>A,B\in\Sigma</math>, then <math>A\cap B, A\cup B, A-B\in\Sigma</math>. (Intersection, union, and difference of two events are events).
| |
| * A '''probability measure''' <math>\Pr:\Sigma\rightarrow\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying
| |
| *:(K3). <math>\Pr(\Omega)=1</math>.
| |
| *:(K4). For any ''disjoint'' events <math>A</math> and <math>B</math> (which means <math>A\cap B=\emptyset</math>), it holds that <math>\Pr(A\cup B)=\Pr(A)+\Pr(B)</math>.
| |
| *:(K5*). For a decreasing sequence of events <math>A_1\supset A_2\supset \cdots\supset A_n\supset\cdots</math> of events with <math>\bigcap_n A_n=\emptyset</math>, it holds that <math>\lim_{n\rightarrow \infty}\Pr(A_n)=0</math>.
| |
| }}
| |
| | |
| ;Remark
| |
| * In general, the set <math>\Omega</math> may be continuous, but we only consider '''discrete''' probability in this lecture, thus we assume that <math>\Omega</math> is either finite or countably infinite.
| |
| * Sometimes it is convenient to assume <math>\Sigma=2^{\Omega}</math>, i.e. the events enumerates all subsets of <math>\Omega</math>. But in general, a probability space is well-defined by any <math>\Sigma</math> satisfying (K1) and (K2). Such <math>\Sigma</math> is called a <math>\sigma</math>-algebra defined on <math>\Omega</math>.
| |
| * The last axiom (K5*) is redundant if <math>\Sigma</math> is finite, thus it is only essential when there are infinitely many events. The role of axiom (K5*) in probability theory is like [http://en.wikipedia.org/wiki/Zorn's_lemma Zorn's Lemma] (or equivalently the [http://en.wikipedia.org/wiki/Axiom_of_choice Axiom of Choice]) in axiomatic set theory.
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| | |
| Useful laws for probability can be deduced from the ''axioms'' (K1)-(K5).
| |
| {{Theorem|Proposition|
| |
| # Let <math>\bar{A}=\Omega\setminus A</math>. It holds that <math>\Pr(\bar{A})=1-\Pr(A)</math>.
| |
| # If <math>A\subseteq B</math> then <math>\Pr(A)\le\Pr(B)</math>. | |
| }}
| |
| {{Proof|
| |
| # The events <math>\bar{A}</math> and <math>A</math> are disjoint and <math>\bar{A}\cup A=\Omega</math>. Due to Axiom (K4) and (K3), <math>\Pr(\bar{A})+\Pr(A)=\Pr(\Omega)=1</math>.
| |
| # The events <math>A</math> and <math>B\setminus A</math> are disjoint and <math>A\cup(B\setminus A)=B</math> since <math>A\subseteq B</math>. Due to Axiom (K4), <math>\Pr(A)+\Pr(B\setminus A)=\Pr(B)</math>, thus <math>\Pr(A)\le\Pr(B)</math>.
| |
| }}
| |
| | |
| ;Notation | |
| An event <math>A\subseteq\Omega</math> can be represented as <math>A=\{a\in\Omega\mid \mathcal{E}(a)\}</math> with a predicate <math>\mathcal{E}</math>.
| |
| | |
| The predicate notation of probability is
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| :<math>\Pr[\mathcal{E}]=\Pr(\{a\in\Omega\mid \mathcal{E}(a)\})</math>.
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| We use the two notations interchangeably.
| |
| | |
| ==Union bound==
| |
| A very useful inequality in probability is the '''Boole's inequality''', mostly known by its nickname '''union bound'''.
| |
| {{Theorem
| |
| |Theorem (union bound)| | |
| :Let <math>A_1, A_2, \ldots, A_n</math> be <math>n</math> events. Then
| |
| ::<math>\begin{align}
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| \Pr\left(\bigcup_{1\le i\le n}A_i\right)
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| &\le
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| \sum_{i=1}^n\Pr(A_i).
| |
| \end{align}</math>
| |
| }}
| |
| {{Proof|
| |
| Let <math>B_1=A_1</math> and for <math>i>1</math>, let <math>B_i=A_i\setminus \left(\bigcup_{j<i}A_j\right)</math>.
| |
| We have <math>\bigcup_{1\le i\le n} A_i=\bigcup_{1\le i\le n} B_i</math>.
| |
| | |
| On the other hand, <math>B_1,B_2,\ldots,B_n</math> are disjoint, which implies by the axiom of probability space that
| |
| :<math>\Pr\left(\bigcup_{1\le i\le n}A_i\right)=\Pr\left(\bigcup_{1\le i\le n}B_i\right)=\sum_{i=1}^n\Pr(B_i)</math>. | |
| Also note that <math>B_i\subseteq A_i</math> for all <math>1\le i\le n</math>, thus <math>\Pr(B_i)\le \Pr(A_i)</math> for all <math>1\le i\le n</math>. The theorem follows.
| |
| }}
| |
| | |
| The union bound is a special case of the '''Boole-Bonferroni inequality'''.
| |
| {{Theorem
| |
| |Theorem (Boole-Bonferroni inequality)|
| |
| :Let <math>A_1, A_2, \ldots, A_n</math> be <math>n</math> events. For <math>1\le k\le n</math>, define <math>S_k=\sum_{i_1<i_2<\cdots<i_k}\Pr\left(\bigcap_{j=1}^k A_{i_j}\right)</math>.
| |
| | |
| :Then for '''''odd''''' <math>m</math> in <math>\{1,2,\ldots, n\}</math>:
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| ::<math>\Pr\left(\bigcup_{1\le i\le n}A_i\right)\le \sum_{k=1}^m (-1)^{k-1} S_k</math>;
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| :and for '''''even''''' <math>m</math> in <math>\{1,2,\ldots, n\}</math>:
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| ::<math>\Pr\left(\bigcup_{1\le i\le n}A_i\right)\ge \sum_{k=1}^m (-1)^{k-1} S_k</math>.
| |
| }}
| |
| The inequality follows from the well-known '''inclusion-exclusion principle''', stated as follows, as well as the fact that the quantity <math>S_k</math> is ''unimodal'' in <math>k</math>.
| |
| {{Theorem
| |
| |Principle of Inclusion-Exclusion|
| |
| :Let <math>A_1, A_2, \ldots, A_n</math> be <math>n</math> events. Then
| |
| ::<math>\Pr\left(\bigcup_{1\le i\le n}A_i\right)=\sum_{k=1}^n (-1)^{k-1} S_k,</math>
| |
| :where <math>S_k=\sum_{i_1<i_2<\cdots<i_k}\Pr\left(\bigcap_{j=1}^k A_{i_j}\right)</math>.
| |
| }}
| |
| | |
| = Conditional Probability =
| |
| In probability theory, the word "condition" is a verb. "Conditioning on the event ..." means that it is assumed that the event occurs.
| |
| | |
| {{Theorem
| |
| |Definition (conditional probability)|
| |
| :The '''conditional probability''' that event <math>A</math> occurs given that event <math>B</math> occurs is
| |
| ::<math>
| |
| \Pr[A\mid B]=\frac{\Pr[A\wedge B]}{\Pr[B]}.
| |
| </math>
| |
| }}
| |
| | |
| The conditional probability is well-defined only if <math>\Pr[B]\neq0</math>.
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| | |
| == Law of total probability ==
| |
| The following fact is known as the law of total probability. It computes the probability by averaging over all possible cases.
| |
| {{Theorem
| |
| |Theorem (law of total probability)| | |
| :Let <math>B_1,B_2,\ldots,B_n</math> be ''mutually disjoint'' events, and <math>\bigcup_{i=1}^n B_i=\Omega</math> is the sample space.
| |
| :Then for any event <math>A</math>,
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| ::<math>
| |
| \Pr[A]=\sum_{i=1}^n\Pr[A\wedge B_i]=\sum_{i=1}^n\Pr[A\mid B_i]\cdot\Pr[B_i].
| |
| </math>
| |
| }}
| |
| {{Proof| Since <math>B_1,B_2,\ldots, B_n</math> are mutually disjoint and <math>\bigvee_{i=1}^n B_i=\Omega</math>, events <math>A\wedge B_1, A\wedge B_2,\ldots, A\wedge B_n</math> are also mutually disjoint, and <math>A=\bigcup_{i=1}^n\left(A\cap B_i\right)</math>. Then the additivity of disjoint events, we have
| |
| :<math>
| |
| \Pr[A]=\sum_{i=1}^n\Pr[A\wedge B_i]=\sum_{i=1}^n\Pr[A\mid B_i]\cdot\Pr[B_i].
| |
| </math>
| |
| }}
| |
| | |
| The law of total probability provides us a standard tool for breaking a probability into sub-cases. Sometimes this will help the analysis.
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|
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|
| == "The Chain Rule" == | | |header1 =Instructor |
| By the definition of conditional probability, <math>\Pr[A\mid B]=\frac{\Pr[A\wedge B]}{\Pr[B]}</math>. Thus, <math>\Pr[A\wedge B] =\Pr[B]\cdot\Pr[A\mid B]</math>. This hints us that we can compute the probability of the AND of events by conditional probabilities. Formally, we have the following theorem:
| | |label1 = |
| {{Theorem|Theorem|
| | |data1 = |
| :Let <math>A_1, A_2, \ldots, A_n</math> be any <math>n</math> events. Then
| | |header2 = |
| ::<math>\begin{align} | | |label2 = |
| \Pr\left[\bigwedge_{i=1}^n A_i\right]
| | |data2 = 张洋 |
| &=
| | |header3 = |
| \prod_{k=1}^n\Pr\left[A_k \mid \bigwedge_{i<k} A_i\right].
| | |label3 = Email |
| \end{align}</math>
| | |data3 = yangzhang@nju.edu.cn |
| }}
| | |header4 = |
| {{Proof|It holds that <math>\Pr[A\wedge B] =\Pr[B]\cdot\Pr[A\mid B]</math>. Thus, let <math>A=A_n</math> and <math>B=A_1\wedge A_2\wedge\cdots\wedge A_{n-1}</math>, then | | |label4= office |
| :<math>\begin{align}
| | |data4= 仙林大气楼 B410 |
| \Pr[A_1\wedge A_2\wedge\cdots\wedge A_n]
| | |header5 = Class |
| &=
| | |label5 = |
| \Pr[A_1\wedge A_2\wedge\cdots\wedge A_{n-1}]\cdot\Pr\left[A_n\mid \bigwedge_{i<n}A_i\right].
| | |data5 = |
| \end{align}
| | |header6 = |
| </math>
| | |label6 = Class meetings |
| Recursively applying this equation to <math>\Pr[A_1\wedge A_2\wedge\cdots\wedge A_{n-1}]</math> until there is only <math>A_1</math> left, the theorem is proved.
| | |data6 = 周四 下午 2:00-4:00，线上课程腾讯会议号：5966863228 |
| | |header7 = |
| | |label7 = Place |
| | |data7 = |
| | |header8 = |
| | |label8 = Office hours |
| | |data8 = 周四 下午4:00-4:30 <br>线上答疑 |
| | |header9 = Reference book |
| | |label9 = |
| | |data9 = |
| | |header10 = |
| | |label10 = |
| | |data10 = {{Infobox |
| | |name = |
| | |bodystyle = |
| | |title = |
| | |titlestyle = |
| | |image = [[File:James-Circulating.jpg|border|100px]] |
| | |imagestyle = |
| | |caption = Introduction to Circulating Atmospheres, <br>''I. James'', Cambridge Press, 1995 |
| | |captionstyle = }} |
| | |header11 = |
| | |label11 = |
| | |data11 = {{Infobox |
| | |name = |
| | |bodystyle = |
| | |title = |
| | |titlestyle = |
| | |image = [[File:Oort.jpg|border|100px]] |
| | |imagestyle = |
| | |caption =Physics of Climate, ''Peixoto, J. P.'' and ''A. H. Oort'', Springer-Verlag New York, 1992 |
| | |captionstyle = |
| }} | | }} |
| | | |header12 = |
| =Random Variable= | | |label12 = |
| {{Theorem|Definition (random variable)| | | |data12 = {{Infobox |
| :A random variable <math>X</math> on a sample space <math>\Omega</math> is a real-valued function <math>X:\Omega\rightarrow\mathbb{R}</math>. A random variable X is called a '''discrete''' random variable if its range is finite or countably infinite. | | |name = |
| | |bodystyle = |
| | |title = |
| | |titlestyle = |
| | |image = [[File:geoff.jpg|border|100px]] |
| | |imagestyle = |
| | |caption =Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, ''Vallis, G. K.'', Cambridge University Press, 2006 |
| | |captionstyle = |
| }} | | }} |
| | | |belowstyle = background:#ddf; |
| For a random variable <math>X</math> and a real value <math>x\in\mathbb{R}</math>, we write "<math>X=x</math>" for the event <math>\{a\in\Omega\mid X(a)=x\}</math>, and denote the probability of the event by
| | |below = |
| :<math>\Pr[X=x]=\Pr(\{a\in\Omega\mid X(a)=x\})</math>.
| |
| | |
| The independence can also be defined for variables:
| |
| {{Theorem
| |
| |Definition (Independent variables)| | |
| :Two random variables <math>X</math> and <math>Y</math> are '''independent''' if and only if
| |
| ::<math>
| |
| \Pr[(X=x)\wedge(Y=y)]=\Pr[X=x]\cdot\Pr[Y=y]
| |
| </math>
| |
| :for all values <math>x</math> and <math>y</math>. Random variables <math>X_1, X_2, \ldots, X_n</math> are '''mutually independent''' if and only if, for any subset <math>I\subseteq\{1,2,\ldots,n\}</math> and any values <math>x_i</math>, where <math>i\in I</math>,
| |
| ::<math>\begin{align}
| |
| \Pr\left[\bigwedge_{i\in I}(X_i=x_i)\right]
| |
| &=
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| \prod_{i\in I}\Pr[X_i=x_i].
| |
| \end{align}</math>
| |
| }} | | }} |
|
| |
|
| Note that in probability theory, the "mutual independence" is <font color="red">not</font> equivalent with "pair-wise independence", which we will learn in the future.
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|
| |
| = Linearity of Expectation =
| |
| Let <math>X</math> be a discrete '''random variable'''. The expectation of <math>X</math> is defined as follows.
| |
| {{Theorem
| |
| |Definition (Expectation)|
| |
| :The '''expectation''' of a discrete random variable <math>X</math>, denoted by <math>\mathbf{E}[X]</math>, is given by
| |
| ::<math>\begin{align}
| |
| \mathbf{E}[X] &= \sum_{x}x\Pr[X=x],
| |
| \end{align}</math>
| |
| :where the summation is over all values <math>x</math> in the range of <math>X</math>.
| |
| }}
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|
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|
| Perhaps the most useful property of expectation is its '''linearity'''.
| | This is the page for the class ''General Circulation of the Atmosphere (大气环流)'' for the Fall 2021 semester. Students who take this class should check this page periodically for content updates and new announcements. |
|
| |
|
| {{Theorem
| | = Announcement = |
| |Theorem (Linearity of Expectations)|
| | * 由于疫情影响，本学期前段大气环流课采取线上直播课的形式， <font color="red" size="2>直播软件为腾讯会议，课程会议号为：5966863228；课程交流QQ群：924139004 </font>。【2021.9.2】 |
| :For any discrete random variables <math>X_1, X_2, \ldots, X_n</math>, and any real constants <math>a_1, a_2, \ldots, a_n</math>,
| |
| ::<math>\begin{align}
| |
| \mathbf{E}\left[\sum_{i=1}^n a_iX_i\right] &= \sum_{i=1}^n a_i\cdot\mathbf{E}[X_i].
| |
| \end{align}</math>
| |
| }}
| |
| {{Proof| By the definition of the expectations, it is easy to verify that (try to prove by yourself):
| |
| for any discrete random variables <math>X</math> and <math>Y</math>, and any real constant <math>c</math>,
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| * <math>\mathbf{E}[X+Y]=\mathbf{E}[X]+\mathbf{E}[Y]</math>; | |
| * <math>\mathbf{E}[cX]=c\mathbf{E}[X]</math>.
| |
| The theorem follows by induction.
| |
| }}
| |
| The linearity of expectation gives an easy way to compute the expectation of a random variable if the variable can be written as a sum.
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|
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|
| ;Example
| | = Course info = |
| : Supposed that we have a biased coin that the probability of HEADs is <math>p</math>. Flipping the coin for n times, what is the expectation of number of HEADs? | | * '''Instructor ''': 张洋， |
| : It looks straightforward that it must be np, but how can we prove it? Surely we can apply the definition of expectation to compute the expectation with brute force. A more convenient way is by the linearity of expectations: Let <math>X_i</math> indicate whether the <math>i</math>-th flip is HEADs. Then <math>\mathbf{E}[X_i]=1\cdot p+0\cdot(1-p)=p</math>, and the total number of HEADs after n flips is <math>X=\sum_{i=1}^{n}X_i</math>. Applying the linearity of expectation, the expected number of HEADs is: | | :*office: 仙林气象楼 B410 |
| ::<math>\mathbf{E}[X]=\mathbf{E}\left[\sum_{i=1}^{n}X_i\right]=\sum_{i=1}^{n}\mathbf{E}[X_i]=np</math>. | | :*email: yangzhang@nju.edu.cn |
| | * '''Class meeting''': 周四 下午 2:00-4:00 |
| | * '''Office hour''': 周四 <font color="red" size="2>下午4:00-4:30，线上答疑</font> |
| | * '''Prerequisites''': 动力气象，天气学，气候学 |
| | * '''Grading''': 平时作业（50％）＋ 期末考试（50％） |
| | 本课程将大致布置4次作业，每次作业一二道题目左右。题目将选择每个课题最具有代表性、需要一定思维强度和动手能力的训练用题目,意在使学生通过顺利完成作业来建立环流系统的物理模型、以对课程内容得到深刻全面地理解和掌握。期末考试题目数量将会比平时作业多,覆盖面更广,但会比作业题目简单,只涉及对基本内容的掌握和对环流理论的直接应用。 |
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| The real power of the linearity of expectations is that it does not require the random variables to be independent, thus can be applied to any set of random variables. For example:
| | = Course Slides = |
| :<math>\mathbf{E}\left[\alpha X+\beta X^2+\gamma X^3\right] = \alpha\cdot\mathbf{E}[X]+\beta\cdot\mathbf{E}\left[X^2\right]+\gamma\cdot\mathbf{E}\left[X^3\right].</math>
| | ''Many figures in the course slides are adapted from the reference books, NOAA and other educational sources. These figures are <font color="red" size="2">for class use only.</font>'' |
|
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| However, do not exaggerate this power!
| | = Assignments = |
| * For an arbitrary function <math>f</math> (not necessarily linear), the equation <math>\mathbf{E}[f(X)]=f(\mathbf{E}[X])</math> does <font color="red">not</font> hold generally.
| | # [[Assignment 1, Fall 2021|Reanalysis data and the earth's climatology]] [Due:2021.10.28] <font color="red" size="1.5">请在本次作业的截止日期前将本次作业的图片部分和文字部分打包作为附件发送至邮箱 circulation_nju@126.com </font> |
| * For variances, the equation <math>var(X+Y)=var(X)+var(Y)</math> does <font color="red">not</font> hold without further assumption of the independence of <math>X</math> and <math>Y</math>.
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|
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|
| ==Conditional Expectation == | | = Course intro = |
| | “大气环流”常指地球大气较大空间范围、较长时间尺度上的空气流动，及其对地球大气热量、动量、能量和水汽的全球输送。虽然从十七、十八世纪起人们就开始研究大尺度的大气运动(如Hadley在1735年提出的信风理论)，但大气环流真正发展成为一门较完备的学科方向却是近半个世纪的事情。随着四五十年代探空资料等高空气象要素的取得，以及六十年代卫星等覆盖全球的观测资料的加入，大气环流的空间结构和时间变化开始被系统、全面地揭示。与此同时，大气环流的数值模拟，也开始成为研究大气环流的一个主要方法，并发展至今成为了解和预估未来气候变化的主要手段。随着观测和模拟手段的进步，大气环流的理论研究也在近三十年开始快速地发展，人们对各种环流系统的维持和变化有了更全面、更深刻、也更为现代的理解。 |
|
| |
|
| Conditional expectation can be accordingly defined:
| | 现代的大气环流是大气动力学、天气学和气候学相结合的产物。大气环流，既是各种天气现象产生的背景流场，又是各种气候状态形成的动力机制。大气环流在低频、季节、年际、年代际等时间尺度的变化，不但会引起天气现象的变化，也影响着气候状态的形成。而在大气科学领域面临着诸如全球暖化、气候变化、环流异常等重大科学问题的今天，大气环流研究的重要性被推到了前所未有的高度,大气环流也成为活跃发展又充满挑战的学科方向。 |
| {{Theorem
| |
| |Definition (conditional expectation)|
| |
| :For random variables <math>X</math> and <math>Y</math>,
| |
| ::<math>
| |
| \mathbf{E}[X\mid Y=y]=\sum_{x}x\Pr[X=x\mid Y=y],
| |
| </math>
| |
| :where the summation is taken over the range of <math>X</math>.
| |
| }}
| |
|
| |
|
| There is also a '''law of total expectation'''.
| | 本课程将讲述在过去几十年里大气环流在观测、理论和模拟上取得的进展。希望学生借此课程能熟悉大气环流的基本分布和形态，掌握各主要环流系统的维持和变化机制，建立各环流系统形成的物理模型，了解现阶段的大气环流模式，知道大气环流方向有待解决的科学问题。 |
| {{Theorem
| |
| |Theorem (law of total expectation)|
| |
| :Let <math>X</math> and <math>Y</math> be two random variables. Then
| |
| ::<math>
| |
| \mathbf{E}[X]=\sum_{y}\mathbf{E}[X\mid Y=y]\cdot\Pr[Y=y].
| |
| </math>
| |
| }}
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|
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| = <math>k</math>-wise independence =
| | 作为一门课程，大气环流内容的讲述常可以有两条线索。一条是全球尺度上大气热量、动量、能量和水汽的分布与输送，Lorenz（1967）和 Peixoto and Oort（1992）是按此线索介绍大气环流的优秀教材；另一条线索，是各纬度、各区域内大气环流系统的形成、维持和变化机制，James（1995）和 Vallis（2006）是按此线索介绍大气环流的经典讲义。根据现阶段大气环流方向的研究特点，本课程的讲述将主要按照后一种方式来展开，并辅以介绍各环流系统对大气各要素场的输送。在介绍各环流系统时，本课程将以观测、理论和模拟为顺序，从各大气环流系统的观测事实入手，介绍大气环流系统的分布特征和时空变化特征；着重介绍关于环流系统的各种动力学模型和现阶段对环流系统的理解；辅以对环流系统模拟研究的介绍；最后通过三者的对比，讨论各环流系统有待研究的问题。 |
| Recall the definition of independence between events:
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| {{Theorem
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| |Definition (Independent events)|
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| :Events <math>\mathcal{E}_1, \mathcal{E}_2, \ldots, \mathcal{E}_n</math> are '''mutually independent''' if, for any subset <math>I\subseteq\{1,2,\ldots,n\}</math>,
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| ::<math>\begin{align}
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| \Pr\left[\bigwedge_{i\in I}\mathcal{E}_i\right]
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| &=
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| \prod_{i\in I}\Pr[\mathcal{E}_i].
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| \end{align}</math>
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| }}
| |
| Similarly, we can define independence between random variables:
| |
| {{Theorem
| |
| |Definition (Independent variables)|
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| :Random variables <math>X_1, X_2, \ldots, X_n</math> are '''mutually independent''' if, for any subset <math>I\subseteq\{1,2,\ldots,n\}</math> and any values <math>x_i</math>, where <math>i\in I</math>,
| |
| ::<math>\begin{align}
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| \Pr\left[\bigwedge_{i\in I}(X_i=x_i)\right]
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| &=
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| \prod_{i\in I}\Pr[X_i=x_i].
| |
| \end{align}</math>
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| }}
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| | |
| Mutual independence is an ideal condition of independence. The limited notion of independence is usually defined by the '''k-wise independence'''.
| |
| {{Theorem
| |
| |Definition (k-wise Independenc)|
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| :1. Events <math>\mathcal{E}_1, \mathcal{E}_2, \ldots, \mathcal{E}_n</math> are '''k-wise independent''' if, for any subset <math>I\subseteq\{1,2,\ldots,n\}</math> with <math>|I|\le k</math>
| |
| :::<math>\begin{align}
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| \Pr\left[\bigwedge_{i\in I}\mathcal{E}_i\right]
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| &=
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| \prod_{i\in I}\Pr[\mathcal{E}_i].
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| \end{align}</math>
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| :2. Random variables <math>X_1, X_2, \ldots, X_n</math> are '''k-wise independent''' if, for any subset <math>I\subseteq\{1,2,\ldots,n\}</math> with <math>|I|\le k</math> and any values <math>x_i</math>, where <math>i\in I</math>,
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| :::<math>\begin{align}
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| \Pr\left[\bigwedge_{i\in I}(X_i=x_i)\right]
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| &=
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| \prod_{i\in I}\Pr[X_i=x_i].
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| \end{align}</math>
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| }}
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| | |
| A very common case is pairwise independence, i.e. the 2-wise independence.
| |
| {{Theorem
| |
| |Definition (pairwise Independent random variables)|
| |
| :Random variables <math>X_1, X_2, \ldots, X_n</math> are '''pairwise independent''' if, for any <math>X_i,X_j</math> where <math>i\neq j</math> and any values <math>a,b</math>
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| :::<math>\begin{align}
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| \Pr\left[X_i=a\wedge X_j=b\right]
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| &=
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| \Pr[X_i=a]\cdot\Pr[X_j=b].
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| \end{align}</math>
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| }}
| |
| | |
| Note that the definition of k-wise independence is hereditary:
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| * If <math>X_1, X_2, \ldots, X_n</math> are k-wise independent, then they are also <math>\ell</math>-wise independent for any <math>\ell<k</math>.
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| * If <math>X_1, X_2, \ldots, X_n</math> are NOT k-wise independent, then they cannot be <math>\ell</math>-wise independent for any <math>\ell>k</math>.
| |
| | |
| == Pairwise Independent Bits ==
| |
| Suppose we have <math>m</math> mutually independent and uniform random bits <math>X_1,\ldots, X_m</math>. We are going to extract <math>n=2^m-1</math> pairwise independent bits from these <math>m</math> mutually independent bits.
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| | |
| Enumerate all the nonempty subsets of <math>\{1,2,\ldots,m\}</math> in some order. Let <math>S_j</math> be the <math>j</math>th subset. Let
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| :<math>
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| Y_j=\bigoplus_{i\in S_j} X_i,
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| </math>
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| where <math>\oplus</math> is the exclusive-or, whose truth table is as follows.
| |
| :{|cellpadding="4" border="1"
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| |-
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| |<math>a</math>
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| |<math>b</math>
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| |<math>a</math><math>\oplus</math><math>b</math>
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| |-
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| | 0 || 0 ||align="center"| 0
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| |-
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| | 0 || 1 ||align="center"| 1
| |
| |-
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| | 1 || 0 ||align="center"| 1
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| |-
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| | 1 || 1 ||align="center"| 0
| |
| |}
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| | |
| There are <math>n=2^m-1</math> such <math>Y_j</math>, because there are <math>2^m-1</math> nonempty subsets of <math>\{1,2,\ldots,m\}</math>. An equivalent definition of <math>Y_j</math> is
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| :<math>Y_j=\left(\sum_{i\in S_j}X_i\right)\bmod 2</math>.
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| Sometimes, <math>Y_j</math> is called the '''parity''' of the bits in <math>S_j</math>.
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| | |
| We claim that <math>Y_j</math> are pairwise independent and uniform.
| |
| | |
| {{Theorem
| |
| |Theorem|
| |
| :For any <math>Y_j</math> and any <math>b\in\{0,1\}</math>,
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| ::<math>\begin{align}
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| \Pr\left[Y_j=b\right]
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| &=
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| \frac{1}{2}.
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| \end{align}</math>
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| :For any <math>Y_j,Y_\ell</math> that <math>j\neq\ell</math> and any <math>a,b\in\{0,1\}</math>,
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| ::<math>\begin{align}
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| \Pr\left[Y_j=a\wedge Y_\ell=b\right]
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| &=
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| \frac{1}{4}.
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| \end{align}</math>
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| }}
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|
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|
| The proof is left for your exercise.
| | = Syllabus = |
| | 本课程具体的内容安排如下：第一章为大气环流的概述，介绍大气环流发展的历史、包含的内容以及大气环流研究的常用观测资料和分析方法。第二章介绍大气环流产生的外部强迫：辐射强迫和下界面过程。第三至六章介绍大气环流中的各个环流系统及它们的动力机制。第七章详细介绍各复杂度的大气环流模式。第八章介绍大气环流领域现阶段最大的一个开放课题：全球暖化背景下的大气环流。这一章既是对前几章所介绍的大气环流理论的应用与检验，又是对未来大气环流研究方向的探讨。借此让学生熟悉并理解大气环流领域亟需解决的课题。具体课程安排和参考书目如下。 |
| | == Course schedule == |
| | *大气环流概述 (Introduction) （4课时） |
| | *大气环流的外部强迫（3课时） |
| | **辐射强迫 (Radiative forcing) |
| | **下界面过程 (Surface boundaries) |
| | *经向环流系统 (Zonally-averaged circulations) |
| | **Hadley 环流（4课时） |
| | **Ferrel 环流，急流，中纬度的波流相互作用（8课时） |
| | *纬向环流系统（Non-zonal circulations）（6课时） |
| | **Storm tracks |
| | **Monsoon |
| | **ENSO and Walker circulation |
| | *能量和水汽循环 (Angular momentum, energy and water vapor)（2课时） |
| | *不同复杂度的大气环流模式 (General circulation in a hierarchy of models)（2课时） |
| | *全球暖化背景下的大气环流 (General circulation in the global warming scenario)（2课时） |
|
| |
|
| Therefore, we extract exponentially many pairwise independent uniform random bits from a sequence of mutually independent uniform random bits.
| | [[点击此处看详细课程安排 (click for more)]] |
|
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|
| Note that <math>Y_j</math> are not 3-wise independent. For example, consider the subsets <math>S_1=\{1\},S_2=\{2\},S_3=\{1,2\}</math> and the corresponding random bits <math>Y_1,Y_2,Y_3</math>. Any two of <math>Y_1,Y_2,Y_3</math> would decide the value of the third one.
| | == References == |
| | *观测部分：Peixoto, J. P. and A. H. Oort, 1992: Physics of Climate. Springer-Verlag New York, Inc., 520 pp. 中文译本：气候物理学，1995，吴国雄、刘辉等译校，气象出版社。 |
| | *综合介绍：James, I., 1995: Introduction to circulating atmospheres. Cambridge University Press, 448 pp. |
| | *理论部分：Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press. 745 pp. |
| | * [[其它参考书目（点击看详情）]] |
