"Probability theory or probability calculus is the branch of mathematics concerned with probability."
The study of random events and processes, including the computation and analysis of probabilities.
Basic Set Theory: A fundamental concept of probability theory which deals with the notion of a collection of objects.
Combinatorics: The study of counting objects that satisfy certain conditions.
Permutations and Combinations: The arrangements and selections of objects with and without repetition.
Probability Axioms: The fundamental mathematical concepts that define probability theory.
Conditional Probability: The probability of an event occurring given that another event has already occurred.
Bayes' Theorem: A formula that describes the probability of an event based on prior knowledge of conditions that might be related to the event.
Random Variables: A variable that takes on a set of possible values with probabilities associated with them.
Probability Distributions: The probability of the occurrence of certain values of a random variable.
Expected Value: The expected outcome of a random variable over a large number of trials.
Law of Large Numbers: A theorem that describes the convergence of the sample mean to the expected value as the number of observations increases.
Central Limit Theorem: A theorem describing the distribution of the sum of a large number of independent, identically distributed random variables.
Hypothesis Testing: The process of determining whether a hypothesis about a population parameter is true or not based on data.
Confidence Intervals: A range of values that is likely to contain an unknown population parameter.
Statistical Inference: The process of drawing conclusions about a population based on a sample.
Markov Chains: A mathematical model that describes a sequence of events where the probability of each event depends only on the state attained in the previous event.
Classical Probability: Refers to the probability calculations based on equally likely outcomes or events.
Empirical Probability: Refers to the probability calculations based on experiments or observations with real data.
Subjective Probability: Refers to the probability calculations based on personal opinions or beliefs.
Conditional Probability: Refers to the probability of an event A given the occurrence of another event B.
Joint Probability: Refers to the probability of two or more events occurring together.
Marginal Probability: Refers to the probability of an event occurring independently of other events.
Bayes' Theorem: Refers to the probability calculations that enables updated probability estimates after new information that describes the state of the system is revealed.
Markov Chain: Refers to the probability modelling technique that depends only on the current state of a system and not on its past history.
Poisson Processes: Refers to the probability calculations of estimating the number of events occurring in a certain time or space interval.
Random Variables: Refers to the probability calculations that describe the numerical outcomes of an experiment with a certain probability distribution.
Stochastic Processes: Refers to the probability calculations that involve the analysis of random processes over time or space.
Large Deviations Theory: Refers to the probability calculations that estimate the tail probabilities of probability distributions. It is concerned with the probability of events that are rare but inevitable, such as the probability of a large stock market crash.
Information Theory: Refers to a branch of probability that estimates the amount of information required to describe an event or system, and its transmission through a communication channel.
Game Theory: Refers to the mathematical study of decision-making under uncertainty and strategic behavior in multi-agent systems.
Measure Theory: Refers to the mathematical tools used to formalize the notion of probability in abstract spaces, and to define probability distributions rigorously.
Bayesian Networks: Refers to the graphical representation of probabilistic relationships among variables, and the systematic updating of these relationships based on new data.
Queuing Theory: Refers to the probability calculations that models the behavior of systems in which requests arrive randomly and must be processed by limited resources, such as computer networks or traffic intersections.
"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."