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## Probability theory

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Showing Rating details. All Languages. More filters. Sort order. Danzu Pakon rated it it was amazing May 05, This course is available with permission as an outside option to students on other programmes where regulations permit. The course is offered as a regular examinable half-unit as well as a service to students and academic staff. Analysis and algebra at the level of a BSc in pure or applied mathematics and basic statistics and probability theory with stochastic processes.

Knowledge of measure theory is not required as the course gives a self-contained introduction to this branch of analysis.

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The course covers core topics in measure theoretic probability and modern stochastic calculus, thus laying a rigorous foundation for studies in statistics, actuarial science, financial mathematics, economics, and other areas where uncertainty is essential and needs to be described with advanced probability models. Emphasis is on probability theory as such rather than on special models occurring in its applications. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov.

Kolmogorov combined the notion of sample space , introduced by Richard von Mises , and measure theory and presented his axiom system for probability theory in This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.

Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more. Consider an experiment that can produce a number of outcomes.

The set of all outcomes is called the sample space of the experiment. The power set of the sample space or equivalently, the event space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number.

These collections are called events. If the results that actually occur fall in a given event, that event is said to have occurred. To qualify as a probability distribution , the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events events that contain no common results, e.

### Learning Outcomes

This event encompasses the possibility of any number except five being rolled. When doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable. A random variable is a function that assigns to each elementary event in the sample space a real number.

This function is usually denoted by a capital letter. This does not always work.

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For example, when flipping a coin the two possible outcomes are "heads" and "tails". Discrete probability theory deals with events that occur in countable sample spaces. Examples: Throwing dice , experiments with decks of cards , random walk , and tossing coins. Classical definition : Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. The modern definition does not try to answer how probability mass functions are obtained; instead, it builds a theory that assumes their existence [ citation needed ].

Continuous probability theory deals with events that occur in a continuous sample space.

Classical definition : The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox.

## Basic Probability Theory with Applications | Mario Lefebvre | Springer

That is, F x returns the probability that X will be less than or equal to x. Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables including discrete random variables that take values in R. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.

Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space :.

The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theoretic approach free of fallacies. For example, to study Brownian motion , probability is defined on a space of functions. When it's convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure.

Discrete densities are usually defined as this derivative with respect to a counting measure over the set of all possible outcomes. Densities for absolutely continuous distributions are usually defined as this derivative with respect to the Lebesgue measure. If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions. Certain random variables occur very often in probability theory because they well describe many natural or physical processes.