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Course# COM-417: Advanced probability and applications

Summary

In this course, various aspects of probability theory are considered. The first part is devoted to the main theorems in the field (law of large numbers, central limit theorem, concentration inequalities), while the second part focuses on the theory of martingales in discrete time.

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Instructors (2)

Related MOOCs (38)

Lectures in this course (75)

Related courses (132)

Olivier Lévêque

Olivier Lévêque was born in Switzerland in 1971. He received the physics diploma from EPFL in 1995 and completed his PhD in mathematics at EPFL in 2001. Since then, he has been with the Laboratory of Information Theory at EPFL. He spent the academical year 2005-2006 at the Electrical Engineering Department of Stanford University, where he was appointed as lecturer. His research interests include stochastic analysis, random matrices, wireless communications and information theory.

Advanced statistical physics

We explore statistical physics in both classical and open quantum systems. Additionally, we will cover probabilistic data analysis that is extremely useful in many applications.

Advanced statistical physics

We explore statistical physics in both classical and open quantum systems. Additionally, we will cover probabilistic data analysis that is extremely useful in many applications.

Algebra (part 1)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Expectation: Basic Properties

Discusses integrability, square-integrability, boundedness, centering, and linearity of random variables.

Martingale Convergence Theorem

Explains the martingale convergence theorem and its applications in probability theory.

Conditional Expectation: Definition

Explains conditional expectation, conditioning events, probabilities calculation, and examples with dice and random variables.

Martingale Convergence Theorem

Explores the proof of the martingale convergence theorem and the conditions for convergence to a random variable.

Independence of Sub-Fields

Explores the concept of independence of sub-fields within a field and its implications in random variables.

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a

MATH-432: Probability theory

The course is based on Durrett's text book
Probability: Theory and Examples.

It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.

It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.

COM-406: Foundations of Data Science

We discuss a set of topics that are important for the understanding of modern data science but that are typically not taught in an introductory ML course. In particular we discuss fundamental ideas an

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A basic course in probability and statistics

Related concepts (339)

Optional stopping theorem

In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e.

Concentration inequality

In probability theory, concentration inequalities provide bounds on how a random variable deviates from some value (typically, its expected value). The law of large numbers of classical probability theory states that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results show that such behavior is shared by other functions of independent random variables.

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Mars rover

A Mars rover is a motor vehicle designed to travel on the surface of Mars. Rovers have several advantages over stationary landers: they examine more territory, they can be directed to interesting features, they can place themselves in sunny positions to weather winter months, and they can advance the knowledge of how to perform very remote robotic vehicle control. They serve a different purpose than orbital spacecraft like Mars Reconnaissance Orbiter. A more recent development is the Mars helicopter.

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".