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Concept in probability theory and statistics
complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may
Complex_random_variable
Variable representing a random phenomenon
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which
Random_variable
Statistical distribution of complex random variables
{N}}_{\mathcal {C}}} , characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters:
Complex_normal_distribution
and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a
Complex_random_vector
Mathematical technique
In statistics, the algebra of random variables provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into
Algebra_of_random_variables
Topics referred to by the same term
Inappropriateness, conduct seen as unethical Immorality Obscenity Vulgarity A complex random variable not being proper Appearance of impropriety, a legal–ethical concept
Impropriety
Probability that random variable X is less than or equal to x
statistics, the cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle
Cumulative distribution function
Cumulative_distribution_function
Concept in probability theory
is zero. Formally: X,Y uncorrelated⟺E[XY]=E[X]⋅E[Y] Two complex random variables Z,W are called uncorrelated if their covariance KZW=E[(Z−E[Z])(W−E[W])‾]
Uncorrelatedness
Moment of a random variable minus its mean
a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable
Central_moment
Concept in probability and statistics
statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability
Independent and identically distributed random variables
Independent_and_identically_distributed_random_variables
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
Topics referred to by the same term
Propositional variable, taking the value true or false in mathematical logic Random variable, a variable in statistics whose value depends on random events Categorical
Variable
Measure of the joint variability
variability of two random variables. The sign of the covariance shows the tendency in the linear relationship between the variables. Covariance is positive
Covariance
Fourier transform of the probability density function
characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Random variable with multiple component dimensions
probability and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either
Multivariate_random_variable
Tree-based ensemble machine learning methods
𝜣M are independent random variables, distributed as a generic random variable 𝜣, independent of the sample 𝒟n. This random variable can be used to describe
Random_forest
Measure of covariance of components of a random vector
variance of a complex scalar-valued random variable with expected value μ {\displaystyle \mu } is conventionally defined using complex conjugation: var
Covariance_matrix
Probability distribution
is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x)=12πσ2exp(−(x−μ)22σ2)
Normal_distribution
Matrix-valued random variable
mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability
Random_matrix
Power series derived from a discrete probability distribution
discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability
Probability generating function
Probability_generating_function
Description of continuous random distribution
continuous random variable, is a function whose value at any given point in the sample space (the set of possible values taken by the random variable) can be
Probability_density_function
Probability distribution
continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then
Log-normal_distribution
Topics referred to by the same term
Nash equilibrium Proper subset Proper space Proper class Proper complex random variable Proper (liturgy), the part of a Christian liturgy that is specific
Proper
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
Probability distribution
distribution of the ratio of two independent normally distributed random variables with mean zero. The Cauchy distribution is often used in statistics
Cauchy_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
real-valued random variables, each of which clusters around a mean value. The multivariate normal distribution of a k-dimensional random vector X = (
Multivariate normal distribution
Multivariate_normal_distribution
Set of random variables
physics and probability, a Markov random field (MRF), Markov network or undirected graphical model is a set of random variables having a Markov property described
Markov_random_field
Mathematical inequality explaining concentration of random variables
probability of a random variable deviating from some value (typically, its expected value). The deviation or other function of the random variable can be thought
Concentration_inequality
Apparent lack of pattern or predictability in events
probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow
Randomness
Concept in measure theory
real-valued random variables { X i } i ∈ I {\displaystyle \{X_{i}\}_{i\in I}} is tight if and only if there exists an almost surely finite random variable X {\displaystyle
Tightness_of_measures
Probability distribution
of random variables having two other known distributions. Given two (usually independent) random variables X and Y, the distribution of the random variable
Ratio_distribution
measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach
Weakly_measurable_function
Statistical measure of how far values spread from their average
as the expected value of the squared deviation from the mean of a random variable. The standard deviation is the square root of the variance. Technically
Variance
Type of matrix in probability theory and statistics
covariance matrix. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has
Cross-covariance_matrix
When the occurrence of one event does not affect the likelihood of another
values of the random variables are real numbers. It has the advantage of working also for complex-valued random variables or for random variables taking values
Independence (probability theory)
Independence_(probability_theory)
Probability theory
an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context
Inverse_distribution
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
Statistical study of photon counting
superposed field will be random. Therefore, the complex amplitude β ( t ) {\displaystyle \beta (t)} is a stochastic variable. It represents the sum of
Photon_statistics
Graph generated by a random process
standard random graphs to more complex graph models . Once we have a model of random graphs, every function on graphs, becomes a random variable. The study
Random_graph
Type of signal in signal processing
serially uncorrelated random variables with a mean of zero and a finite variance; a single realization of white noise is a random shock. In some contexts
White_noise
Subset of variables that contains all the useful information
Markov blanket of a random variable is a set of variables that renders the variable conditionally independent of all other variables in the system. This
Markov_blanket
with just the random contributions and therefore, is just the possibility distribution of the random contributions. A random-fuzzy variable can be constructed
Random-fuzzy_variable
Bias in causal inference
be a variable that (1) independently predicts the outcome (or dependent variable), (2) is associated with the exposure (or independent variable), and
Confounding
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)
Type of random mathematical object
distribution is the probability distribution of a random variable N (called a Poisson random variable) such that the probability that N equals n is given
Poisson_point_process
Sampling technique
Simple random sampling is a basic type of sampling and can be a component of other more complex sampling methods. The principle of simple random sampling
Simple_random_sample
experiments, completely randomized designs are for studying the effects of one primary factor without the need to take other nuisance variables into account. This
Completely_randomized_design
Expectation or average of the falling factorial of a random variable
moment of a probability distribution on the real or complex numbers, or, in other words, a random variable X with that probability distribution, is E [ (
Factorial_moment
Averages of repeated trials converge to the expected value
we have obtained By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as
Law_of_large_numbers
Branch of mathematics concerning probability
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide
Probability_theory
On eigenvalues of random matrices
asserts that for any sequence of random n × n matrices whose entries are independent and identically distributed random variables, all with mean zero and variance
Circular_law
convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free
Free_convolution
Type of statistical model
occurrence), non-randomly varying (meaning that the intercepts and/or slopes are predictable from an independent variable at Level 2), or randomly varying (meaning
Multilevel_model
Noncentral generalization of the chi-squared distribution
independent, normally distributed random variables with means μ i {\displaystyle \mu _{i}} and unit variances. Then the random variable ∑ i = 1 k X i 2 {\displaystyle
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
Process forming a path from many random steps
walk formally, take independent random variables Z 1 , Z 2 , … {\displaystyle Z_{1},Z_{2},\dots } , where each variable is either 1 or −1, with a 50% probability
Random_walk
Probability distribution
distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees
Rayleigh_distribution
Class of stochastic process
stationary process where the sample space is also discrete (so that the random variable may take one of N {\displaystyle N} possible values) is a Bernoulli
Stationary_process
Symbol representing a mathematical object
Euclidean geometry or the corresponding axes z for a complex number, or in statistics a normal random variable α, β, γ, θ, φ for angle measures ε (with δ as
Variable_(mathematics)
Distribution of variables which satisfies a stability property under linear combinations
two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be
Stable_distribution
Type of probability distribution
distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance
Mixture_distribution
Type of function
for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable
Singular_function
Structural analysis of a network
In network science, a biased random walk on a graph is a time path process in which an evolving variable jumps from its current state to one of various
Biased_random_walk_on_a_graph
to be the complex plane. In other words, if the random variable Y has a Cauchy distribution with complex parameter θ, then the random variable Y * defined
McCullagh's parametrization of the Cauchy distributions
McCullagh's_parametrization_of_the_Cauchy_distributions
Selection of data points in statistics
Implementation usually follows a simple random sample. In addition to allowing for stratification on an ancillary variable, poststratification can be used to
Sampling_(statistics)
The objective function would be: minx∈θf(x)=minx∈θE[F(x,y)] y is a random variable that represents the noise. x is the parameter that minimizes f(x).
Simulation-based_optimization
Computer simulation with random inputs
system that has variables that can change stochastically (randomly) with individual probabilities. Realizations of these random variables are generated
Stochastic_simulation
Statistical test for causality
to compare ρ¯ to ρS¯ computed from S random realizations (surrogates) of X. CCM is used to detect if two variables belong to the same dynamical system
Convergent_cross_mapping
Theorem of convex functions
probability theory, it is generally stated in the following form: if X is a random variable and φ is a convex function, then φ(E[X])≤E[φ(X)]. The difference
Jensen's_inequality
Overview of and topical guide to probability
Goodman–Nguyen–van Fraassen algebra Discrete random variables: Probability mass functions Continuous random variables: Probability density functions Normalizing
Outline_of_probability
Statistical tool to model changing systems
system with a random variable that changes through time. In this context, the Markov property indicates that the distribution for this variable depends only
Markov_model
Hardware description and hardware verification language
for randomization: pre_randomize, randomize and post_randomize. The randomize method is called by the user for randomization of the class variables. The
SystemVerilog
Variable capable of taking on a limited number of possible values
values of a categorical variable is referred to as a level. The probability distribution associated with a random categorical variable is called a categorical
Categorical_variable
Branch of statistics
encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those
Mathematical_statistics
Probability distribution on complex matrices
independent Gaussian random variables. It has support for p × p {\displaystyle p\times p} Hermitian positive definite matrices. The complex Wishart distribution
Complex_Wishart_distribution
Statistical transform
are described by two independent and normally distributed random variables, the random variables for R2 and Θ (shown above) in the corresponding polar coordinates
Box–Muller_transform
Probabilistic problem-solving algorithm
value of some random variable can be approximated by taking the empirical mean (a.k.a. the 'sample mean') of independent samples of the variable. When the
Monte_Carlo_method
Probabilistic model
which a graph expresses the conditional dependence structure between random variables. Graphical models are commonly used in probability theory, statistics—particularly
Graphical_model
Model in theoretical ecology and statistical mechanics
sets a timescale, and α {\displaystyle \alpha } is a random matrix whose entries are random variables with mean ⟨ α i j ⟩ = μ α / S {\displaystyle \langle
Random generalized Lotka–Volterra model
Random_generalized_Lotka–Volterra_model
Star whose brightness fluctuates, as seen from Earth
A variable star is a star whose brightness as seen from Earth (its apparent magnitude) changes systematically with time. This variation may be caused by
Variable_star
Flaw in mathematical modelling
independent variable is known as the "one in ten rule"). In the process of regression model selection, the mean squared error of the random regression
Overfitting
Concept in gambling, insurance, and finance
exact solution if the random variable added at each step is not a Gaussian random variable but rather a binomial random variable with parameter n=2. However
Risk_of_ruin
Design of tasks
and unknown confounding variables between groups on average. This strengthens causal inference. The risks associated with random allocation (such as having
Design_of_experiments
Concept in mathematics or computer science
science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound
Free variables and bound variables
Free_variables_and_bound_variables
Statistical measure used in survey research
random variable. model based - in this case, the sample is fixed, the weights are fixed, but the outcome of interest is treated as a random variable.
Design_effect
Thematic map visualizing multiple variables
strong correlation between the two variables. If there is no clear pattern, the map can become an overwhelming mix of random symbols. A second problem occurs
Multivariate_map
Branch of statistics
independent variable. Statistical inference is generally used to determine the difference between variations in the original data that are random variation
Causal_inference
Type of digital communication link sharing
slots (of fixed length) or data frames (of variable lengths) that often appear to be scheduled in a randomized order, and experience varying delay (while
Statistical time-division multiplexing
Statistical_time-division_multiplexing
Statistical model
non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random variables. In
Fixed_effects_model
Random matrix with gaussian entries
In random matrix theory, the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled
Gaussian_ensemble
Study of collection and analysis of data
(IID) random variables with a given probability distribution: standard statistical inference and estimation theory defines a random sample as the random vector
Statistics
Mathematical concept
integrability using the notation expectation of random variables., that is, 1. A class 𝒞 of random variables is called uniformly integrable if: There exists
Uniform_integrability
Collection of statistical models
used to describe otherwise complex relations among variables. A dog show provides an example. A dog show is not a random sampling of the breed: it is
Analysis_of_variance
Integral transform useful in probability theory, physics, and engineering
a real variable (usually t {\displaystyle t} , in the time domain) to a function of a complex variable s {\displaystyle s} (in the complex-valued frequency
Laplace_transform
Mathematical transform that expresses a function of time as a function of frequency
characteristic function Φ of the probability density function f of a random variable X of continuous type is defined without a negative sign in the exponential
Fourier_transform
Type of gradient noise in computer graphics
application include procedurally generating terrain, applying pseudo-random changes to a variable, and assisting in the creation of image textures. It can be formulated
Perlin_noise
Experiment methodology
survey data, offline data, and other, more complex phenomena. "A/B testing" is a shorthand for a simple randomized controlled experiment, in which a number
A/B_testing
Scientific procedure performed to validate a hypothesis
involve variables that are difficult to quantify or control. Observational studies are limited because they lack the statistical properties of randomized experiments
Experiment
Optimization method
that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random. Stochastic optimization
Stochastic_optimization
exists for all complex s with re(s) ≥ σ. The Laplace–Stieltjes transform appears naturally in the following context. If X is a random variable with cumulative
Laplace–Stieltjes_transform
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COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
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