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Concept in statistics
identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling. Formally
Exchangeable_random_variables
Concept in probability and statistics
i.i.d. variables are exchangeable random variables, introduced by Bruno de Finetti.[citation needed] Exchangeability means that while variables may not
Independent and identically distributed random variables
Independent_and_identically_distributed_random_variables
Topics referred to by the same term
Look up exchangeable in Wiktionary, the free dictionary. Exchangeable may refer to: Exchangeable batteries, used with battery swapping in charging stations
Exchangeable
Conditional independence of exchangeable observations
the variables of the exchangeable sequence are not themselves independent, only exchangeable, there is an underlying family of i.i.d. random variables. That
De_Finetti's_theorem
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution
Convergence of random variables
Convergence_of_random_variables
Convergence of random variables / (LS:R) Doob's martingale convergence theorems / (SU:R) Ergodic theory / (S:R) Exchangeable random variables / (S:BR) Hewitt–Savage
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Probability distribution
random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable
Distribution of the product of two random variables
Distribution_of_the_product_of_two_random_variables
Disintegration theorem Bayes' theorem de Finetti's theorem Exchangeable random variables Rule of succession Conditional independence Conditional event
List_of_probability_topics
Averages of repeated trials converge to the expected value
Ho Lee (1998). "A Note on the Weak Law of Large Numbers for Exchangeable Random Variables" (PDF). Communications of the Korean Mathematical Society. 13
Law_of_large_numbers
Sigma-algebra used in probability and ergodic theory
an exchangeable event or symmetric event, and the sigma-algebra of invariant events is often called the exchangeable sigma-algebra. A random variable on
Invariant_sigma-algebra
Mathematical problem involving optimal stopping theory
fully formal statement is as below: Does there exist an exchangeable sequence of random variables X 1 , . . . , X n {\displaystyle X_{1},...,X_{n}} , such
Secretary_problem
Function in mathematical analysis
example: If X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} are exchangeable random variables, then the function E ∏ j = 1 n X j a j {\displaystyle {\text{E}}\prod
Schur-convex_function
Probability distribution
are involved, such as Binomial random variables, associated with binary response variables; Poisson random variables, associated with rare events; Thermal
Normal_distribution
Function that is invariant under all permutations of its variables
functions – Functions such that f(–x) equals f(x) or –f(x) Exchangeable random variables – Concept in statistics Quasisymmetric function Ring of symmetric
Symmetric_function
Mathematical function for the probability a given outcome occurs in an experiment
probability. Probability distributions are closely linked to random variables. A random variable is a function that assigns a value to each outcome of a probabilistic
Probability_distribution
Collection of random variables
a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables in a probability space, where the
Stochastic_process
Generalization of the one-dimensional normal distribution to higher dimensions
over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out)
Multivariate normal distribution
Multivariate_normal_distribution
Probability distribution
statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central
Log-normal_distribution
Expected value of a random variable given that certain conditions are known to occur
mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on
Conditional_expectation
Graph partition into regular subgraphs
arXiv:math/0612838, Bibcode:2006math.....12838I Austin, Tim (2008), "On exchangeable random variables and the statistics of large graphs and hypergraphs", Probability
Szemerédi_regularity_lemma
statistical analysis. Developed the representation theorem for exchangeable random variables showing that they are the basis of the IID model in statistics
Founders_of_statistics
Measure of dependence between two variables
the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the
Mutual_information
Discrete probability distribution
dependent random variables with a specific distribution D {\displaystyle D} . Because most of the theorems about bounds in sum of random variables are concerned
Hypergeometric_distribution
statistics Exact test Examples of Markov chains Excess risk Exchange paradox Exchangeable random variables Expander walk sampling Expectation–maximization algorithm
List_of_statistics_articles
Function type in graph theory
(possibly random) graphon. That is, a random graph model has a jointly exchangeable adjacency matrix if and only if it is a jointly exchangeable random graph
Graphon
Bias in causal inference
of a set Z of variables that would guarantee unbiased estimates must be done with caution. The criterion for a proper choice of variables is called the
Confounding
and 30s. Importance: Emphasizes exchangeable random variables which are often mixtures of independent random variables. Argues for finitely additive probability
List of publications in statistics
List_of_publications_in_statistics
Statistical relationship
statistics, correlation is a type of statistical relationship between two random variables or bivariate data. It usually refers to the extent to which a pair
Correlation
Concept in probability theory
inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality
Markov's_inequality
Random process of binary (boolean) random variables
binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi
Bernoulli_process
Class of statistics in estimation theory
independent and identically distributed random variables, or more generally for exchangeable sequences, such as in simple random sampling from a finite population
U-statistic
Divide and conquer sorting algorithm
viewpoint, variables such as lo and hi do not use constant space; it takes O(log n) bits to index into a list of n items. Because there are such variables in
Quicksort
Statistics named for Richard von Mises
Daffer, P.Z.; Patterson, R.F. (1985). Limit theorems for sums of exchangeable random variables. New Jersey: Rowman and Allanheld. von Mises, R. (1947). "On
V-statistic
Bound on probability of a random variable being far from its mean
stated for random variables, but can be generalized to a statement about measure spaces. Let X {\displaystyle X} (integrable) be a random variable with finite
Chebyshev's_inequality
Concept in statistics
statistics, a Gaussian random field (GRF) is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is
Gaussian_random_field
Statistical method in genetic epidemiology
unbiased estimates of the effects of an assumed causal variable without conducting a traditional randomized controlled trial (the standard in epidemiology for
Mendelian_randomization
Theorem in probability theory
}} be a sequence of independent and identically distributed random variables taking values in a set X {\displaystyle \mathbb {X} } . The Hewitt-Savage
Hewitt–Savage_zero–one_law
Computational concept
uniformly random seed, generates a highly random output that appears independent from the source and uniformly distributed. Examples of weakly random sources
Randomness_extractor
Star whose brightness fluctuates, as seen from Earth
or by something partly blocking the light, so variable stars are classified as either: Intrinsic variables, whose inherent luminosity changes; for example
Variable_star
Probability distribution
random variable X ~ B(n, p) can be considered as the sum of n Bernoulli distributed random variables. So the sum of two Binomial distributed random variables
Binomial_distribution
Type of random mathematical object
technique originally developed for approximating random variables such as Gaussian and Poisson variables, which has also been applied to point processes
Poisson_point_process
Sequence where any order is equally likely
A random permutation is a sequence where any order of its items is equally likely at random, that is, it is a permutation-valued random variable of a set
Random_permutation
Stochastic differential equation
and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly
Langevin_equation
Average uncertainty in variable's states
theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible
Entropy_(information_theory)
Descriptive statistic
of a systematic difference. The ICC is constructed to be applied to exchangeable measurements — that is, grouped data in which there is no meaningful
Intraclass_correlation
Statistical regression technique
geographic units with few respondents. For a state-level random effect, for example, the exchangeable prior can be replaced with a l state ∼ N ( γ 0 + γ 1
Multilevel regression with poststratification
Multilevel_regression_with_poststratification
Representation of a type of random process
of random process. It can be used to describe time-varying processes from many natural and artificial sources. The model specifies output variables that
Autoregressive_model
Probability distribution
of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior
Beta_distribution
Statistical model written in multiple levels
y_{2},\ldots } is exchangeable. For any n, the sequence y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable. Bayesian hierarchical
Bayesian hierarchical modeling
Bayesian_hierarchical_modeling
Measure of linear correlation
every random variable has zero mean, and T is the data transformed so all variables have zero mean and zero correlation with all other variables – the
Pearson correlation coefficient
Pearson_correlation_coefficient
Particular case of the generalized extreme value distribution
distribution. This is useful because the difference of two Gumbel-distributed random variables has a logistic distribution. The Gumbel distribution is named after
Gumbel_distribution
Discrete probability distribution
random variable Y ∼ CMB ( n , p , ν ) {\displaystyle Y\sim \operatorname {CMB} (n,p,\nu )} may be written as a sum of exchangeable Bernoulli random
Conway–Maxwell–binomial distribution
Conway–Maxwell–binomial_distribution
Concept in statistics
with (some of) the parameters of that distribution themselves being random variables. If the parameter is a scale parameter, the resulting mixture is also
Compound probability distribution
Compound_probability_distribution
Statistical distribution for dependence between random variables
each variable is uniform on the interval [0, 1]. Copulas are used to describe / model the dependence (inter-correlation) between random variables. Their
Copula_(statistics)
Statistical modeling technique
variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable
Quantile_regression
Probabilistic problem-solving algorithm
simulation. Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different
Monte_Carlo_method
Statistical method
variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible
Factor_analysis
Sampling technique
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals
Simple_random_sample
Bound on optimal stopping in random sequences
J. H. (Ben) Garling. It concerns a process in which a sequence of random variables X i {\displaystyle X_{i}} arrive from known distributions D i {\displaystyle
Prophet_inequality
Random model in mathematics
the weaker property of exchangeability. Recall that a (finite or infinite) sequence of random variables is called exchangeable if its joint distribution
Pólya_urn_model
and correction of differences in coding schemes: variables are compared with coding schemes of variables external to the data set, and possibly corrected
Data_analysis
Exact statistical hypothesis test
the labels are exchangeable under the null hypothesis, then the resulting tests yield exact significance levels; see also exchangeability. Confidence intervals
Permutation_test
Family of stochastic processes
the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular
Dirichlet_process
Method in probability theory
distribution of a sum of m {\displaystyle m} -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform) metric
Stein's_method
Solution to a stochastic differential equation
subjected to random displacements due to collisions with other particles, which is called Brownian motion. The position of the particle is then random; its probability
Diffusion_process
Optimization method
that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random. Stochastic optimization
Stochastic_optimization
Way of inferring information from cross-covariance matrices
X = (X1, ..., Xn) and Y = (Y1, ..., Ym) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find
Canonical_correlation
independent, family of random variables, and let H {\displaystyle \mathbf {H} } be a function that maps n {\displaystyle n} variables to a self-adjoint matrix
Matrix_Chernoff_bound
Regularization technique for ill-posed problems
the coefficients of multiple-regression models in scenarios where the variables are highly correlated. It has been used in many fields including econometrics
Ridge_regression
Memoryless property of a stochastic process
a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network
Markov_property
Probability distribution
integrating out the Dirichlet random variable. This causes the various categorical variables drawn from the same Dirichlet random variable to become correlated
Dirichlet_distribution
Type of quantum mechanics theory
of quantum mechanics by introducing additional, possibly inaccessible, variables. The mathematical formulation of quantum mechanics assumes that the state
Hidden-variable_theory
Algorithm for linear programming
canonical form. The variables corresponding to the columns of the identity matrix are called basic variables while the remaining variables are called nonbasic
Simplex_algorithm
Discrete-time stochastic process
process is described on page 92. Pitman, Jim (1995). "Exchangeable and Partially Exchangeable Random Partitions". Probability Theory and Related Fields.
Chinese_restaurant_process
Malagasy algebraic divination by seeds
peoples in Madagascar. It involves algorithmic operations performed on random data generated from tree seeds, which are ritually arranged in a tableau
Sikidy
Stochastic volatility model used in derivatives markets
F} and σ {\displaystyle \sigma } are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential
SABR_volatility_model
where all linear combinations of coordinates are normally distributed random variables. Gauss–Markov process (cf. below) GenI process Girsanov's theorem Hawkes
List of stochastic processes topics
List_of_stochastic_processes_topics
Statistical concept
optimal e-variables for small blocks of outcomes and these are then multiplied to obtain e-variables for larger samples - these e-variables work well
E-values
Statistical method
and satisfies a subtle pairwise exchangeable condition: for any j {\displaystyle j} , the joint distribution of the random matrix [ X , X ~ ] {\displaystyle
Knockoffs_(statistics)
Body of matter in a state of internal equilibrium
from equilibrium, in addition to constitutive variables that was described above, a set of internal variables ξ 1 , ξ 2 , … {\displaystyle \xi _{1},\xi _{2}
Thermodynamic_system
Greek-American engineer and applied mathematician (1921–2002)
communications, and signal and system theory. His classic book Probability, Random Variables, and Stochastic Processes is used as a textbook in many graduate-level
Athanasios_Papoulis
Computer network management and monitoring protocol
manager-to-agent request to retrieve the value of a variable or list of variables. Desired variables are specified in variable bindings (the value field is not used)
Simple Network Management Protocol
Simple_Network_Management_Protocol
Method of resource allocation
of sd-efficiency, for the same fair random assignment setting but with strict item rankings. Define the exchange graph of a given fractional allocation
Ordinal_Pareto_efficiency
some other variables, implies that such selection bias can be ignored, so one can recover (or estimate) the causal effect. Missing at random Yamamoto,
Ignorability
Real function with secant line between points above the graph itself
expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as
Convex_function
Concept in information theory
maximized for a given variance. A Gaussian random variable has the largest entropy amongst all random variables of equal variance, or, alternatively, the
Differential_entropy
2015 password-based key derivation function
PDF says it's ℋ (but doesn't document what ℋ is). It's actually Blake2b. Variable length items are prepended with their length as 32-bit little-endian integers
Argon2
Method to solve optimization problems
newly introduced slack variables, x {\displaystyle \mathbf {x} } are the decision variables, and z {\displaystyle z} is the variable to be maximized. The
Linear_programming
Methods employed to reduce error in science tests
or observation designed to minimize the influence of variables other than the independent variable under investigation, thereby reducing the risk of confounding
Scientific_control
Algorithm for statistical inference on graphical models
Markov random fields. It calculates the marginal distribution for each unobserved node (or variable), conditional on any observed nodes (or variables). Belief
Belief_propagation
Differential equation containing derivatives with respect to only one variable
independent variable, and, less commonly, in contrast with stochastic differential equations (SDEs) where the modeled process is random. A linear differential
Ordinary differential equation
Ordinary_differential_equation
Set of objects whose state must satisfy limits
maintains a partial assignment of the variables. Initially, all variables are unassigned. At each step, a variable is chosen, and all possible values are
Constraint satisfaction problem
Constraint_satisfaction_problem
Estimate of an interval in which future observations will fall
applies not just to sampling from a population, but to any exchangeable sequence of random variables, not necessarily independent or identically distributed
Prediction_interval
discrete-time processes via a waiting time distribution are called continuous-time random walks. An example of a continuous-time stochastic process for which sample
Continuous-time stochastic process
Continuous-time_stochastic_process
Augmented password-authenticated key exchange protocol
password. A and B are random one time ephemeral keys of the user and host respectively. | (pipe) denotes concatenation. All other variables are defined in terms
Secure Remote Password protocol
Secure_Remote_Password_protocol
Principle used in linear cryptanalysis
more naturally when the random variables take values in { − 1 , 1 } {\displaystyle \{-1,1\}} . If we introduce variables χ i = 1 − 2 X i = ( − 1 )
Piling-up_lemma
Greatest and least values in a statistical data sample
interval: in a sample from a population, or more generally an exchangeable sequence of random variables, each observation is equally likely to be the maximum
Sample_maximum_and_minimum
Differential equations involving stochastic processes
listed company shares, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in
Stochastic differential equation
Stochastic_differential_equation
Statistical hypothesis test
treatment and control within each pair makes the observations exchangeable. For an exchangeable distribution, X i − Y i {\displaystyle X_{i}-Y_{i}} has the
Wilcoxon_signed-rank_test
Generalization of mutual information for more than two variables
(redundancy or synergy) bound up in a set of variables, beyond that which is present in any subset of those variables. Unlike the mutual information, the interaction
Interaction_information
EXCHANGEABLE RANDOM-VARIABLES
EXCHANGEABLE RANDOM-VARIABLES
EXCHANGEABLE RANDOM-VARIABLES
EXCHANGEABLE RANDOM-VARIABLES
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EXCHANGEABLE RANDOM-VARIABLES
EXCHANGEABLE RANDOM-VARIABLES
EXCHANGEABLE RANDOM-VARIABLES
EXCHANGEABLE RANDOM-VARIABLES