"Probability theory or probability calculus is the branch of mathematics concerned with probability."
The branch of mathematics that deals with the study of random events and their properties.
Set theory: The study of sets and their properties.
Combinatorics: The study of counting and arranging objects in different ways.
Permutations and combinations: The ways of arranging or selecting objects.
Probability spaces: The formal definition of probability in a set-based framework.
Random variables: Mathematical models of random events.
Distribution functions: Functions that describe the probability of a random variable taking on certain values.
Probability mass/density functions: Describes the probability distribution of a discrete or continuous random variable.
Conditional probability: The probability of an event given the occurrence of another event.
Bayes' theorem: A theorem that describes the relationship between conditional probabilities.
Independence: Two events are independent if the occurrence of one event does not influence the probability of the occurrence of the other event.
Law of large numbers: As the number of independent and identically distributed trials increases, the sample mean converges to the expected value of the random variable.
Central limit theorem: The distribution of the sample mean in a large population approaches a normal distribution, regardless of the distribution of the population.
Markov chains: Mathematical models of systems that change over time.
Stochastic processes: Random processes that evolve over time.
Poisson process: A type of stochastic process that models the occurrence of rare events over time.
Brownian motion: A continuous-time stochastic process that models the random motion of particles.
Martingales: Mathematical models that describe systems that are fair in the long run.
Information theory: The study of quantifying and transmitting information, including measures such as entropy and mutual information.
Bayesian inference: A probabilistic approach to statistical inference that updates prior probability distributions based on new data.
Monte Carlo methods: A class of computational techniques that use random numbers to simulate and approximate complex systems.
Classical Probability Theory: This is the most basic form of probability theory, where the outcomes of an experiment are equally likely to occur.
Combinatorial Probability theory: This is the study of discrete structures, such as graphs and trees, which are used to analyze various types of combinatorial problems.
Bayesian Probability Theory: This is a probabilistic approach to solving problems, where prior knowledge is used to calculate the probability of future events.
Decision Theory: This is the study of making decisions under uncertainty, where the outcomes of different actions are uncertain.
Stochastic Process Theory: This is a branch of probability theory that deals with the study of random processes, which are used to model many real-world phenomena.
Markov Chain Theory: This is a type of stochastic process theory that deals with the study of Markov chains, which are used to model systems where the future depends only on the present state.
Queueing Theory: This is the study of the behavior of queues, which are used to model many systems, such as call centers and traffic flow.
Martingale Theory: This is a branch of probability theory that deals with the study of martingales, which are used to model various processes, such as stock prices and gambling games.
Extreme Value Theory: This is the study of the distribution of extreme events, such as floods and earthquakes.
"Probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms."
"Typically these axioms formalize probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space."
"Any specified subset of the sample space is called an event."
"Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes."
"Stochastic processes provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion."
"Two major results in probability theory describing such behavior are the law of large numbers and the central limit theorem."
"It is not possible to perfectly predict random events."
"As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data."
"Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation."
"A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics."
"...expressing it through a set of axioms."
"...a measure taking values between 0 and 1."
"A set of outcomes called the sample space."
"Discrete and continuous random variables..."
"...mathematical abstractions of non-deterministic or uncertain processes or measured quantities..."
"The law of large numbers describes the behavior of random events."
"The central limit theorem describes the behavior of random events."
"Probability theory is essential to many human activities that involve quantitative analysis of data."
"Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state."