In Sections 1.1–1.3 we describe three examples of coupling illustrating both the method and its usefulness. ... Probabilistic Methods in Geotechnical Engineering Probabilistic Methods in Geotechnical Engineering 7 which gives the probability of the event in question by using the probabilities that each ‘cause’ occurs. endobj endobj Jan Bouda (FI MU) Lecture 4 - The Probabilistic Method March 27, 2012 4 / 60 The Asymptotic Notation: Little-o Note the di erence between de nition for the big-O notation, and the endobj Probabilistic Lens (Ch 2) 11/5/2016 << /S /GoTo /D (section.3) >> endobj endobj 196 0 obj endobj 56 0 obj << /S /GoTo /D (section.18) >> << /S /GoTo /D (section.13) >> 21 0 obj endobj << /S /GoTo /D (subsection.16.2) >> 3/31/11) endobj endobj 296 0 obj These are the lecture notes for a two-hours lecture, held by us in the summer semester 2018 at Technische Universit at Berlin. Lecture 11 (Feb 10): Lovasz Local Lemma [MU 6.7-6.9] Anna's notes Alistair Sinclair notes: these and these; Moser and Tardos; Tim Roughgarden notes; Terry Tao blog post. endobj << /S /GoTo /D (subsection.23.1) >> 137 0 obj endobj endobj Having it is only a good start, you have to carefully study to learn its content. << /S /GoTo /D (subsection.16.1) >> (Thu. 52 0 obj endobj endobj In addition to being a useful method in its own right, it will also be a stepping stone towards Bayesian modeling. endobj My notes for each lecture are limited to 4 pages. (Dependent Random Choice) It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is strictly greater than zero. (More bounds) If you carry it to the lectures then you don’t have to make notes, you can add things to the margins. Lecture 2 Notes on Probabilistic Method The probability that a vertex is in Y is (since there is probability pthat a given vertex is in X; we care about the vertex and its neighbors) P(v2Y) = (1 p)deg(v)+1 (1 p) +1: Hence E(jYj) n(1 p) +1: (2) Then by linearity of expectation with (1) and (2), E(jUj) = E(jXj+ jYj) = E(jXj) + E(jYj) pn+ (1 p) +1n: 240 0 obj 148 0 obj Week 5 (10/21, 10/23): Metric embeddings continued, and one lecture on the "probabilistic method". endobj 209 0 obj 233 0 obj endobj 293 0 obj << /S /GoTo /D (section.24) >> endobj << /S /GoTo /D (section.8) >> The above file needs 46 pages to print, having two small pages per A4 printer page, and here is a more usual A4 format (ca. This is regarded as a very powerful tool by the researchers working on the theory of differential equations. endobj (Hypergraph coloring) endobj endobj endobj 164 0 obj endobj endobj 205 0 obj 2/3/2011) (Tue. << /S /GoTo /D (section.9) >> Memorial University of Newfoundland. %PDF-1.4 (Eigenvalues and expanders) 12 0 obj 70 pages). Subjects: Max Cut, Ramsey numbers, Hamiltonian paths in tournaments, sperner's thm . 157 0 obj 72 0 obj 268 0 obj 236 0 obj /Length 4252 165 0 obj endobj << /S /GoTo /D (subsection.15.2) >> 2/17/11) 3 0 obj << endobj 121 0 obj 128 0 obj endobj 265 0 obj In case you nd a serious error, please send email to the instructor pointing it out. endobj 213 0 obj << /S /GoTo /D (section.16) >> Probabilistic Method in Engineering. Ch 1, Lecture notes: 3/5/2016: finding the closest vertex in a parallelepiped; maximum number of Hamilton cycles in a tournament. endobj 200 0 obj (Pseudorandomness) endobj << /S /GoTo /D (subsection.3.2) >> endobj endobj Lecture Notes 22 - Probabilistic Method. University. 81 0 obj endobj endobj 276 0 obj endobj (Applications of Talagrand's inequality) (Crossing number, incidences, and sum-product estimates) 2MMD30: Graphs and Algorithms Lecture 8 Date: 9/3/2016 Instructor: Nikhil Bansal 1 Applications of the probabilistic method As will shall see repeatedly in the rest of the course, the probabilistic method is a very powerful tool for proving theorems. 248 0 obj 5/12/11) 36 0 obj 101 0 obj << /S /GoTo /D (subsection.5.1) >> 197 0 obj (Tue. 192 0 obj endobj 124 0 obj (Applications of FKG inequality) 232 0 obj 177 0 obj 225 0 obj << /S /GoTo /D (subsection.5.2) >> 108 0 obj (Thu. endobj 96 0 obj Lecture 11: The probabilistic method Instructor: Jacob Fox Very often, we need to construct a combinatorial object satisfying properties, for example to show a counterexample or a lower bound for a certain statement. 304 0 obj 25 0 obj 80 0 obj 84 0 obj (Random Graphs) 212 0 obj << /S /GoTo /D (subsection.8.3) >> 4/5/11) The lecture notes are based on Jeremy Bradley’s lecture from 2014/15, but the course has been partially restructured since then. << /S /GoTo /D (section.17) >> endobj endobj 152 0 obj endobj (Tue. endobj endobj endobj 141 0 obj (Tue. endobj 13 0 obj endobj (Clique Number) endobj 224 0 obj << /S /GoTo /D (section.2) >> I will be away on … Each of these examples will be worked out in more detail later. 289 0 obj This is called the probabilistic method. (Tue. 57 0 obj Lecture 1. In this lecture, we will study some basic principles of the probabilistic method, a combinatorial tool with many applications in computer science. 301 0 obj Each section focuses on a different technique, along with examples of applications. 24 0 obj (Thu. endobj endobj << /S /GoTo /D (subsection.3.1) >> (Thu. << /S /GoTo /D (subsection.14.1) >> 140 0 obj (Tue. endobj endobj 264 0 obj Lecture notes - Chapter 02 - Advanced Natural Gas Engineering - Probabilistic Method In Engineering. 1 0 obj 201 0 obj 237 0 obj 221 0 obj 41 0 obj endobj endobj 112 0 obj Lecture notes Lecture 8. We introduce the basic idea through several examples drawn from early lecture notes, and follow that by 169 0 obj If you decide not to print it endobj endobj << /S /GoTo /D (subsection.18.2) >> 92 0 obj The basic recipe for applying the probabilistic method … Note that T is a stopping time, i.e., for each n∈N0 the event << /S /GoTo /D (subsection.12.1) >> << /S /GoTo /D (subsection.1.2) >> (Tue. endobj >> << /S /GoTo /D (subsection.4.2) >> 76 0 obj 284 0 obj (Some bounds) endobj Advanced Natural Gas Engineering (ENGI … endobj (A general setting) << /S /GoTo /D (subsection.24.1) >> (Quadratic residue tournament) 188 0 obj (Tue. << /S /GoTo /D (subsection.6.1) >> 4/26/11) endobj << /S /GoTo /D (subsection.2.3) >> endobj << /S /GoTo /D (subsection.26.1) >> 149 0 obj endobj (Weierstrass Approximation Theorem) 4/7/11) << /S /GoTo /D (subsection.20.1) >> << /S /GoTo /D (subsection.7.1) >> << /S /GoTo /D (section.11) >> 2/24/11) (Ramsey Numbers) endobj endobj endobj endobj 4/12/11) However, these books were not generally available to students in Prague, and this was the main reason for starting with the present notes. endobj Random walks and electrical networks. Lectures on the Probabilistic Method In first semester 2013 I will give some lectures on the Probabilistic Method of Paul Erdős. endobj << /S /GoTo /D (section.22) >> endobj << /S /GoTo /D (subsection.2.1) >> 208 0 obj 2/10/11) endobj (Combinatorial Geometry) (Sum-free subsets) The Probabilistic Method in Combinatorics Lecturer: Yufei Zhao Notes by: Andrew Lin Spring 2019 This is an edited transcript of the lectures of MIT’s Spring 2019 class 18.218: The Probabilistic Method in Combinatorics, taught by Professor Yufei Zhao. endobj << /S /GoTo /D (subsection.10.1) >> 3/17/11) A Special Note: These notes are only an approximation to the actual course. (Tue. 133 0 obj endobj 4/21/11) << /S /GoTo /D (section.7) >> << /S /GoTo /D (subsection.9.2) >> << /S /GoTo /D (section.21) >> endobj << /S /GoTo /D (section.23) >> endobj << /S /GoTo /D (subsection.13.1) >> endobj endobj endobj 77 0 obj 249 0 obj Probabilistic methods – Lecture notes References are for the book ”The Probabilistic Method” by Alon and Spencer, Second Edition. << /S /GoTo /D (subsection.5.3) >> In particular, the split into specific days should be regarded only as an estimate. (Correlation inequalities) So this is the Probabilistic method lecture note. endobj 89 0 obj << /S /GoTo /D (subsection.12.2) >> endobj endobj lecture notes is to give an introduction into the probabilistic method and the in-volved techniques, where we have a preference for elegant solutions rather than intricate calculations. endobj 9 0 obj 125 0 obj 281 0 obj << /S /GoTo /D (subsection.1.1) >> (Ramsey multiplicity) endobj << /S /GoTo /D (subsection.9.1) >> 64 0 obj Note that Lis a function of the model parameters (in this case, ), not the observed data. endobj << /S /GoTo /D (subsection.1.3) >> (Thu. endobj (Martingales and tight concentration) endobj 32 0 obj endobj Lecture Notes Course Home Syllabus Calendar ... Introduction to the Probabilistic Method (PDF) 2–4: Linearity of Expectation (PDF) 5: Alterations (PDF) 6–8: The Second Moment Method (PDF) 9: The Chernoff Bound (PDF) 10–13: 61 0 obj (Tue. endobj << /S /GoTo /D (subsection.15.1) >> (Thu. Course. 184 0 obj (Discrepancy) Course. Alon, Spencer, and Erdős (1992) describe the method as follows: In order to prove the existence of a combinatorial structure with certain properties, we construct an appropriate probability space and show that a randomly chosen element in the space has the desired … << /S /GoTo /D (section.20) >> endobj 2/8/2011) The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. 28 0 obj The “probabilistic method” is a powerful tool in graph theory and combinatorics. 172 0 obj (Chernoff bounds) 277 0 obj – JS The Probabilistic Method. University. << /S /GoTo /D (subsection.6.2) >> 40 0 obj 185 0 obj >> (Chromatic number) •Let X be a random variable. endobj (Independent Sets) endobj 204 0 obj Lecture 15: Learning probabilistic models Roger Grosse and Nitish Srivastava 1 Overview ... the full Bayesian method, and the maximum a-posteriori approximation. << /S /GoTo /D (subsection.4.3) >> 2/1/2011) 3/3/11) %PDF-1.4 (Thu. Explained very basically, this method proves the existence of a configuration, first, by creating a probability space whose points are configurations, and second, by showing a positive probability that the “random configuration” meets the desired criteria. endobj endobj endobj endobj endobj (Compactness arguments) endobj 256 0 obj endobj (Talagrand's Inequality) endobj I suggest you printing it out and carry with yourself to the lectures. A brief introduction to the probabilistic method, followed by some basic bounds and a few examples. 37 0 obj xڽWMs�6��W�Δ0> ���q��v���N�ۜ��";Ω������F'Q�x����$w I��Ȟ���O�'��. << /S /GoTo /D (section.19) >> 120 0 obj (Local lemma on hypergraphs) x��[�o丑�_�C7�fDJ�� ��n��n�,p;s�C��[n+���J�up��)�M���q70`IE���*�w.~�'m/u�ꬾ�ps��eOV��6_}���U����z����_�������J�u�o7�׶\5��v��ow�~�[�d?���lf~�Ka�Fs���7�t�~z��z|��Ԁ�������З�i�3+Uf����○��3��\d���T^��/p�)]ח����r��x�_a�+�*�;�Fu���p����UM��K{=��*W��n}e�l�ss�������XP�Հ�c�q�vd�j����p��2���l>b��q��5w�{`#p�*�]@v0�Z��"���)Te��BAg`��Ʈ~����B�6|�k��i��)wL�u�D��v��c���Abj��G�(�m}7���5����#��Lr� �@o�|\_��ii�k��ײ�����n��CM,c���$�04�}�j��Cr�ݿi�q݁;g��NX�5�\ق����ͨ&[E�t��\��cXf�*b�+O��F��F�oiP+�Hx'��vB�9H�[wU������JF��U��ҽkd�� S�%�k�s-��:�%��.+��'y'$���. (Unbalancing lights) endobj 260 0 obj 3/10/11) 189 0 obj The lecture was indeed based on these. endobj If an object chosen randomly from the universe satisfies a property with positive probability, there must be an object of the universe satisfying that property. << /S /GoTo /D (section.6) >> 104 0 obj << /S /GoTo /D [306 0 R /Fit ] >> Let’s begin with a simple example: we have ipped a particular coin 100 times, and it landed heads N H = 55 times and tails N T = 45 times. 68 0 obj 88 0 obj endobj endobj << /S /GoTo /D (section.10) >> The general idea is to use one of the following two facts. endobj << /S /GoTo /D (subsection.10.2) >> 20 0 obj endobj endobj 33 0 obj << /S /GoTo /D (subsection.21.1) >> 53 0 obj The Probabilistic Method Lecture Notes @inproceedings{Matousek2009ThePM, title={The Probabilistic Method Lecture Notes}, author={J. Matousek and J. Vondr{\'a}k}, year={2009} } J. Matousek, J. Vondrák; Published 2009; If you find errors, please let us know! (Thu. endobj endobj /Length 1065 << /S /GoTo /D (subsection.4.1) >> Time and place The lectures will (usually) be held on Thursdays, from 3pm to 4pm, in Lecture Theatre V101 at the University of Newcastle, Callaghan campus (coordinates D4 on the campus map), starting on Thursday 7 March.. 5/5/11) 3/29/11) << /S /GoTo /D (subsection.18.1) >> << /S /GoTo /D (subsection.19.1) >> These lecture notes are intended for undergraduate students of computer science at Imperial College London and cover basic mathematical concepts that are required in other courses. endobj 216 0 obj (Second Moment Method) 176 0 obj endobj (Number Theory) endobj (Thu. (Four-function theorem) %���� endobj 217 0 obj 280 0 obj endobj endobj 161 0 obj endobj endobj endobj stream 5 0 obj 2/15/11) 132 0 obj 168 0 obj (FKG inequality) 60 0 obj 273 0 obj CS 252, Lecture 10: The Probabilistic Method 1 Introduction Consider the following puzzle: Suppose that 12% of earth’s surface is land, and the rest is water1.Irre- 297 0 obj (Local Coloring) (Thu. Probabilistic Analysis Lecture #10: The Probabilistic Method Gregory Valiant October 30, 2019 1 Introduction The probabilistic method is a surprisingly effective approach for proving that certain combinatorial objects exist. endobj I also include some entertaining, but nonexaminable topics, some of which are unusual for a course at this level (such as random permutations, entropy, re ection principle, Benford and Zipf distributions, Erd}os’s probabilistic method, value at risk, eigenvalues (Tournaments) << /S /GoTo /D (subsection.11.1) >> DOWNLOAD Probabilistic Methods in Applied Physics Lecture Notes in Physics From Springer PDF Online. 144 0 obj These objects could be graphs, hypergraphs, directed graphs, vectors, subsets with a given property, and so on. << /S /GoTo /D (subsection.7.2) >> 5/3/11) Lecture 12 - Probabilistic Method, Cheeger Inequlity, Random Walks1 In this lecture we wrap up our unit on spectral graph theory and begin our discussion of random walks ... 1 These lecture notes are a work in progress and there may be yptos, … endobj endobj 16 0 obj 300 0 obj << /S /GoTo /D (section.1) >> << /S /GoTo /D (subsection.24.2) >> endobj 65 0 obj However, as the topic demands expertise on both PDE and probability theory, an initiative to teach the topic as a structured course is vastly absent globally, including in India. endobj (Thu. endobj The rst method we’ll cover for tting probabilistic models is maximum likelihood. endobj This method uses the fact that if E(X) is the expected value of the << /S /GoTo /D (section.15) >> 44 0 obj << /S /GoTo /D (subsection.8.1) >> endobj endobj endobj endobj 113 0 obj endobj (Max-cut) 49 0 obj (Balancing vectors) Basic notions in probability: expectation, independence. (Quasi-random graphs) << /S /GoTo /D (section.4) >> (Distinct sums) endobj << /S /GoTo /D (subsection.13.2) >> 153 0 obj (Antichains) << /S /GoTo /D (section.12) >> 156 0 obj endobj 180 0 obj 29 0 obj For students, the notes may have another advantage too: 85 0 obj << /S /GoTo /D (section.5) >> (Discrepancy) endobj Although the proof uses … (Linearity of Expectation) 181 0 obj endobj endobj 3/15/11) endobj 4/28/11) 93 0 obj endobj /Filter /FlateDecode 257 0 obj Maximal number of Hamilton paths in tournaments. endobj (Lov\341sz Local Lemma) (Thu. endobj endobj 261 0 obj 100 0 obj (Dependent Random Choice) 116 0 obj << /S /GoTo /D (subsection.2.2) >> endobj endobj /Filter /FlateDecode 105 0 obj Probabilistic method in PDE is equally used in Pure and Applied Mathematics research. 5/10/11) (Erd\366s-Ko-Rado Theorem) 48 0 obj 73 0 obj 97 0 obj 220 0 obj 17 0 obj There is some extra material here, and there will almost certainly be some material in the course not covered in the notes. << /S /GoTo /D (section.25) >> 253 0 obj endobj endobj endobj (Tue. << /S /GoTo /D (subsection.25.3) >> endobj No class (President's Day) (Feb 15) Lecture 12 (Feb 17): Markov chains and random walks. (Graph exposure martingales) 252 0 obj 229 0 obj Engineering Notes and BPUT previous year questions for B.Tech in CSE, Mechanical, Electrical, Electronics, Civil available for free download in PDF format at lecturenotes.in, Engineering Class handwritten notes, exam notes, previous year questions, PDF free download (Tue. 173 0 obj 285 0 obj 292 0 obj 3/8/11) endobj << /S /GoTo /D (subsection.17.1) >> 4 0 obj TU Eindhoven Advanced Algorithms (2IL45) — Course Notes 4.2 The probabilistic method Randomization can not only be used to obtain simple and efficient algorithms, it can also be used in combinatorial proofs. << /S /GoTo /D (section.26) >> (Ramsey Numbers) endobj 2 159 (Discrete Probabilistic Methods) Lecture Notes This course is about an idea pioneered by Erd˝os: to use probability to construct, find, or show the existence of certain discrete objects with a desired property. N. Alon and J. Spencer: The Probabilistic Method, J. Wiley and Sons, New York, NY, 2nd edition, 2000. << /S /GoTo /D (subsection.25.2) >> Carleton University. Lecture 2. (Thu. 305 0 obj 241 0 obj 4/14/11) (Alterations) (Independence number of triangle-free graphs) The existence of tournaments where every set of size k is beaten by somebody (Theorem 1.2.1). 129 0 obj endobj << /S /GoTo /D (subsection.25.1) >> Introduction to Metric Embeddings, and Bourgain's low-distortion randomized embedding of any finite metric into L1. 288 0 obj 109 0 obj 245 0 obj 8 0 obj 136 0 obj (Dominating sets) (Thu. 3/1/11) 69 0 obj Disclaimer: These lecture notes are informal in nature and are not thoroughly proofread. 117 0 obj endobj 45 0 obj 193 0 obj 244 0 obj endobj This method is a powerful tool for demonstrating the existence of combinatorial objects. << /S /GoTo /D (subsection.22.1) >> Sec 2.1, Probabilistic Lens (Ch 4) 10/5/2016: proof of Bregman's Theorem. 272 0 obj In particular we will rarely care about the exact constants in order to keep the exposition as clean as possible. stream << /S /GoTo /D (subsection.8.2) >> << /S /GoTo /D (section.14) >> endobj endobj endobj endobj The probabilistic method (with Jan Vondrak) Lecture notes to a course on the probabilistic, mostly follows the Alon-Spencer book, includes a review of notions from probability theory. (Alterations: Ramsey Numbers) We introduce mathematical concepts and theories for the rigorous probabilistic analysis of a certain type of a telecommunication system, an … The probabilistic method comprises two ideas: Any random variable assumes at least one value not smaller than its expectation. 228 0 obj 145 0 obj 375 0 obj << 160 0 obj 269 0 obj endobj << /S /GoTo /D (subsection.3.3) >> endobj
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