Probability and Statistics

Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology.

put the theory of mathematical probability on a sound footing

is a distance function defined between probability distributions on a given metric space

In the stochastic calculus, Tanaka's formula for the Brownian motion states that

is a statistical measure of process capability

Yuri Linnik was a Ukrainian-born mathematician who worked on number theory, probability theory and mathematical statistics

Heritability is a statistic used in the fields of breeding and genetics that estimates the degree of variation in a phenotypic trait in a population that is due to genetic variation between individuals in that population.

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution o

In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distrib

In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some l

In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) f

In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of prob

In probability theory, Kolmogorov's three-series theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure con

In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the

In probability theory, Azuma's inequality or the Azuma-Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) give

In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard an

In probability theory, the law of large numbers is a mathematical law which states that the average of the results obtained from a

In probability theory, the coupon collector's problem refers to mathematical analysis of "collect all coupons and win" contests.

In statistics, the delta method is a method of deriving the asymptotic distribution of a random variable.

In statistics, Bessel's correction is the use of n − 1 instead of n in the formula for the sample variance and sample standard dev

In statistics, the Fisher transformation (or Fisher z-transformation) of a Pearson correlation coefficient is its inverse hyperbol

In mathematics and statistics, the arithmetic mean ( arr-ith-MET-ik), arithmetic average, or just the mean or average is the sum o

In statistics, the probit function converts a probability (a number between 0 and 1) into a score.

In probability theory, the total variation distance is a statistical distance between probability distributions, and is sometimes

In probability theory and statistics, covariance is a measure of the joint variability of two random variables.

In probability theory and statistics, the zeta distribution is a discrete probability distribution.

In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of

In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMS

In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data.

In statistics, the Bhattacharyya distance is a quantity which represents a notion of similarity between two probability distributi

In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less t

In statistics, Spearman's rank correlation coefficient or Spearman's ρ is a number ranging from −1 to 1 that indicates how strongl

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals

In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (

In statistics, the logit (logistic unit) or log-odds function is the quantile function associated with the standard logistic distr

In statistics and in probability theory, distance correlation is a measure of dependence between two paired random vectors of arbi

In statistics, a zero-inflated model is a statistical model based on a zero-inflated probability distribution, i.e.

In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree centr

In probability theory, an empirical process is a stochastic process that characterizes the deviation of the empirical distribution

In descriptive statistics, the range of a set of data is the size or width of the narrowest interval which contains all the data.

In statistics, the Pearson correlation coefficient (PCC), also known as Pearson's r, the Pearson product-moment correlation coeffi

In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or th

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or sque

In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the sq

In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible e

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probab

In probability theory and statistics, the Weibull distribution is a continuous probability distribution.

In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent rando

In probability theory, Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under ch

In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the sca

In probability theory, Wald's equation, Wald's identity or Wald's lemma is an important identity that simplifies the calculation o

In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its m

In probability theory, Le Cam's theorem, named after Lucien Le Cam, states the following.

In statistics, Basu's theorem states that any boundedly complete and sufficient statistic is independent of any ancillary statisti

In statistics, Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produc

In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality

In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are

In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian mo

modern statistical science

mathematical statistics

statistics and probability

probability theory

statistics

statistician and probabilist

probability theory

probability theory

measure-theoretic probability

mathematician and statistician

He has been credited with establishing the discipline of mathematical statistics.

was a prominent statistician

was a British statistician

was a British statistician

He was elected a fellow of the American Statistical Association in 1975.

However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.

He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics.

Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory.

Kiyosi Itô (伊藤 清, Itō Kiyoshi; Japanese pronunciation: [itoː kiꜜjoɕi], 7 September 1915-10 November 2008) was a Japanese mathematician who made fundamental contributions to probability theory, in particular, the theory of stochastic processes.

Abraham de Moivre (French pronunciation: [abʁaam də mwavʁ]; 26 May 1667-27 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

Pierre-Simon, Marquis de Laplace (; French: [pjɛʁ simɔ̃ laplas]; 23 March 1749-5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy.

In statistics, he pioneered small-sample experimental design.

Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров, IPA: [ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf] , 25 April 1903-20 October 1987) was a Soviet mathematician who played a central role in the creation of modern probability theory.

Calyampudi Radhakrishna Rao (10 September 1920-22 August 2023) was an Indian-American mathematician and statistician.

Egon Sharpe Pearson (11 August 1895-12 June 1980) was an English mathematician and statistician.

Thomas Bayes ( BAYZ; c. 1701-7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem.

He extended foundational results-such as the law of large numbers and the central limit theorem-to sequences of dependent random variables, laying the groundwork for what would become known as Markov chains.

This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations.

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution.

In the mathematical theory of probability, the entropy rate or source information rate of a stochastic process is, informally, the time density of the average information in a stochastic process.

A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose.

In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval.

Mixed logit is a fully general statistical model for examining discrete choices.

In econometrics, the autoregressive conditional heteroskedasticity (ARCH) model is a statistical model for time series data that describes the variance of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms; often the variance is related to the squares of the previous innovations.

It is a summary statistic of statistical dispersion or variability.

In mathematics, Itô's lemma or Itô's formula is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process.

In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero.

Condorcet's jury theorem is a statistical theorem about the probability of a given group of individuals arriving at a correct decision.

Probability and Statistics Rank Statistics

Probability and Statistics Rank Statistics

Probability and Statistics Probability

Probability and Statistics Time-Series Analysis

Probability and Statistics Statistical Distributions

Probability and Statistics Random Walks

Probability and Statistics Statistical Distributions

Probability and Statistics Random Walks

gives information on the asymptotic distribution of a statistic φ(Zε) of a family of random variables Zε as ε

Fuchs, states that for a particle undergoing a zero-mean random walk in m-dimensions, it is certain to come back

in the alternative. Additive measures of belief such as probability do not protect an analyst from making such an

In econometrics, cointegration is a statistical property that describes a long-run equilibrium

by George Edward Pelham Box and Mervin Edgar Muller, is a random number sampling method for generating pairs of

known as the product limit estimator, is a non-parametric statistic used to estimate the survival function from lifetime

strictly proper scoring rule that measures the accuracy of probabilistic predictions. For unidimensional predictions, it is strictly

Aleksandr Yakovlevich Khinchin was a Russian mathematician who contributed to many fields including number theory and probability.

Alfréd Rényi was a Hungarian mathematician who worked in probability theory, combinatorics, graph theory and number theory.

Edward Nelson was an American mathematician who won the Steele Prize for applying probability to quantum field theory. He wrote several classic texts on the topic. In the latter part of his career he studied mathematical logic.

Herbert Robbins was an American mathematician and statistician who worked in topology, measure theory, statistics,and other fields.

The Marquis de Condorcet 's most important work was on probability and the philosophy of mathematics.

Richard Von Mises was an applied mathematician who worked on fluid mechanics, aerodynamics, aeronautics statistics and probability theory. He gave the first university course on powered flight in 1913 .

Sergei Bernstein was a Ukrainian mathematician who made important contributions to partial differential equations, differential geometry, probability theory and approximation theory.

Paul Lévy was a French mathematician who, after starting out as an expert on functional analysis, made important advances in probability theory.