"Probability theory or probability calculus is the branch of mathematics concerned with probability."
Deals with random events and their probabilistic properties.
Set Theory: The study of sets and their properties, used as the foundation for probability theory.
Counting Principles: Techniques for counting number of outcomes, such as permutations, combinations and partitions.
Probability basics: Probability as a measure of uncertainty, independent and dependent events, Bayes’ theorem, conditional probability, and concepts such as expected value and variance.
Discrete Probability Distributions: Probability distribution for discrete random variables, including binomial, Poisson and geometric distributions.
Continuous Probability Distributions: Probability distribution for continuous random variables, including normal, exponential and gamma distributions.
Estimation: Methods for estimating population parameters using sample data, including point estimation, interval estimation, and hypothesis testing.
Regression Analysis: Statistical methods for modeling the relationship between variables, including linear regression, machine learning algorithms, and outlier detection.
Inferential Statistics: Techniques for making inferences about a population based on sample data, including methods such as t-tests and chi-square tests.
Data Visualization: Techniques for visually representing data, including histograms, box plots, scatter plots, and time series plots.
Experimental Design: The design, execution, and analysis of experiments, including factors, treatments, and randomization.
Bayesian Statistics: The theory and methods of Bayesian inference, including the use of prior beliefs to update probabilities and the concept of posterior probability.
Multivariate Analysis: Techniques for analyzing data with more than one variable, including methods such as principal component analysis, cluster analysis, and factor analysis.
Statistical Programming: Programming languages and software tools commonly used in statistical analysis, including R, SAS, MATLAB, and Python.
Stochastic Processes: Mathematical models for random phenomena, including Markov Chains, Brownian Motion and Queuing Theory.
Computer Simulations: Methods for generating and analyzing data using computer models, including Monte Carlo methods and statistical simulations.
Classical probability: The branch of probability theory that deals with situations in which all possible outcomes are equally likely.
Bayesian probability: A branch of probability that takes prior information and beliefs into account when making predictions.
Combinatorial probability: The branch of probability that deals with counting techniques, such as permutations and combinations, that are used to calculate the probability of certain events.
Conditional probability: The probability of an event given that another event has already occurred.
Markov chains: A stochastic process in which the future state of a system depends only on the present state, and not on any past states.
Stochastic processes: A general term for any process that involves randomness.
Monte Carlo methods: A family of computational algorithms that rely on repeated sampling to obtain numerical results.
Statistical inference: The process of using data to make inferences about a population, based on a sample.
Regression analysis: The statistical analysis of relationships between variables, typically employed to make predictions about future behaviour.
Hypothesis testing: The process of testing a hypothesis about a population, based on data from a sample.
Multivariate analysis: The statistical analysis of multiple variables simultaneously.
Time series analysis: The statistical analysis of data that is collected over time.
Non-parametric statistics: A branch of statistics that makes no assumptions about the underlying distribution of data.
Spatial statistics: The statistical analysis of spatial data, typically used in geographical and environmental sciences.
Survival analysis: The analysis of time-to-event data, often used in medical studies.
"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."