List of probability topics

List of Probability Topics

Probability theory is a branch of mathematics concerned with analyzing random phenomena. The fundamental objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. This article aims to provide a comprehensive list of topics related to probability theory, each of which plays a crucial role in understanding the behavior of systems affected by randomness.

Basic Concepts

• Probability Space: The mathematical construct that models a random experiment. It consists of a sample space, a set of events, and a probability measure.
• Random Variable: A variable whose possible values are numerical outcomes of a random phenomenon.
• Probability Distribution: Describes how probabilities are distributed over the values of the random variable.
• Expected Value: The weighted average of all possible values that a random variable can take on.
• Variance and Standard Deviation: Measures of how spread out the values of a random variable are.
• Law of Large Numbers: A theorem that describes the result of performing the same experiment a large number of times.
• Central Limit Theorem: States that, under certain conditions, the sum of a large number of random variables will be approximately normally distributed.

Probability Distributions

• Binomial Distribution: The probability distribution of the number of successes in a sequence of independent experiments.
• Poisson Distribution: A discrete frequency distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.
• Normal Distribution (Gaussian distribution): A very common continuous probability distribution - symmetric and bell-shaped.
• Exponential Distribution: The probability distribution that describes the time between events in a Poisson point process.
• Gamma Distribution: A two-parameter family of continuous probability distributions.

Stochastic Processes

• Markov Chains: Models of random processes that undergo transitions from one state to another on a state space.
• Brownian Motion: A continuous-time stochastic process used to model random movement.
• Poisson Process: A stochastic process that models a series of events occurring at random points in time.
• Random Walk: A mathematical formalization of a path that consists of a succession of random steps.

Advanced Topics

• Bayesian Probability: Interpretation of probability where probability expresses a degree of belief in an event.
• Monte Carlo Methods: A broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.
• Stochastic Differential Equations: Differential equations in which one or more of the terms is a stochastic process.
• Queueing Theory: The mathematical study of waiting lines, or queues.

Applications

Probability theory is applied in a wide variety of fields, including Statistics, Finance, Gambling, Science, Engineering, and Information Technology.