"Probability theory or probability calculus is the branch of mathematics concerned with probability."
Statistics and probability are essential for analyzing and interpreting data. A strong understanding of probability distributions, statistical inference, and hypothesis testing is crucial.
Probability Theory: The foundational understanding of probability and random variables, including probability rules, discrete and continuous probability distributions, conditional probability, and Bayes Theorem.
Statistical Inference: The process of making informed conclusions based on statistical data, including hypothesis testing, confidence intervals, and probability distributions for estimating unknown parameters.
Descriptive Statistics: Techniques for summarizing and describing data, including measures of central tendency and variability, histograms, and scatter plots.
Regression and Correlation Analysis: Techniques for modeling and analyzing relationships between variables, including simple and multiple linear regression, logistic regression, and correlation analysis.
Bayesian Statistics: An alternative approach to statistical inference that incorporates subjective prior beliefs and updates them with data to make posterior probability statements.
Time Series Analysis: Methods for analyzing data that are taken over time, including trend analysis, seasonal variation, and auto-regressive models.
Data Mining and Machine Learning: Techniques for discovering patterns and relationships in large datasets, including cluster analysis, decision trees, and neural networks.
Biostatistics: Statistical concepts and methods specific to the field of biology, including experimental design, survival analysis, and meta-analysis.
Markov Chain Monte Carlo Methods: A powerful class of algorithms for simulating complex systems with probabilistic dependencies, including Gibbs sampling and Metropolis-Hastings sampling.
Spatial Statistics: Techniques for analyzing and modeling data that vary over space, including spatial autocorrelation, kriging, and geostatistics.
Multivariate Analysis: Techniques for analyzing data with multiple variables, including principal component analysis, factor analysis, and discriminant analysis.
Stochastic Processes: Mathematical models for describing random phenomena that evolve over time, including Poisson processes, Brownian motion, and Markov chains.
Descriptive statistics: Describes basic features of data, such as mean, median, mode, range, standard deviation, and probability distribution.
Bayesian statistics: A method of statistical inference in which Bayes' theorem is used to update the probabilities of a hypothesis based on new data.
Frequentist statistics: A school of thought in statistics that emphasizes the use of probability theory to describe the behavior of sample data.
Nonparametric statistics: A branch of statistics that does not assume a particular probability distribution or population parameter but instead uses methods that are more general and less dependent on assumptions.
Time-series analysis: A technique used to analyze data that changes over time, such as gene-expression or protein data.
Survival analysis: A statistical method of analyzing data that relates to time-to-event data, such as the time-to-death of a patient or the time-to-failure of a machine.
Statistical learning: A set of methods and algorithms that are used to analyze large datasets and extract patterns and relationships.
Network analysis: A technique used to analyze complex relationships between genes, proteins, and other biological entities.
High-throughput data analysis: A statistical method of analyzing data from technologies such as RNA-seq, proteomics, and metabolomics.
Computational genetics: A branch of computational biology that uses statistical methods to study the genetic basis of diseases and traits.
Stochastic models: A set of mathematical models that describe the randomness inherent in biological systems, such as gene regulation or metabolic networks.
Simulation methods: A technique that uses computer models to simulate biological processes and generate synthetic data for hypothesis testing.
"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."