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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
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
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
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
Concept in probability theory
In probability theory and statistics, two real-valued random variables, X {\displaystyle X} , Y {\displaystyle Y} , are said to be uncorrelated if their
Uncorrelatedness (probability theory)
Uncorrelatedness_(probability_theory)
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
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
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
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
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
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)
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
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 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
Tree-based ensemble machine learning methods
modern practice of random forests, in particular: Using out-of-bag error as an estimate of the generalization error. Measuring variable importance through
Random_forest
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
is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f ( x ) =
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
Graph generated by a random process
random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random
Random_graph
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
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
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)
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
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
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 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
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
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
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
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
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
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
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
Type 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
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
e., both systematic and random contributions to the total uncertainty. Random-fuzzy variable (RFV) is a type 2 fuzzy variable, defined using the mathematical
Random-fuzzy_variable
Type of random mathematical object
distribution is the probability distribution of a random variable N {\textstyle N} (called a Poisson random variable) such that the probability that N {\displaystyle
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
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
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
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
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
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
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
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)
Process of making something random
data to the broader population. Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes
Randomization
On constructing objects that obey a system of constraints with limited dependence
An} are determined by a finite collection of mutually independent random variables, a simple Las Vegas algorithm with expected polynomial runtime proposed
Algorithmic Lovász local lemma
Algorithmic_Lovász_local_lemma
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
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
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
Design of tasks
more independent variables, also referred to as "input variables" or "predictor variables." The change in one or more independent variables is generally hypothesized
Design_of_experiments
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
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
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
Correlation of a signal with a time-shifted copy of itself, as a function of shift
Essentially, it quantifies the similarity between observations of a random variable at different points in its domain (commonly, time). The analysis of
Autocorrelation
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
Computer security technique
Address space layout randomization (ASLR) is a computer security technique involved in preventing exploitation of memory corruption vulnerabilities. In
Address space layout randomization
Address_space_layout_randomization
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
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
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
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
Airplane wings capable of changing position to alter their geometry
A variable-sweep wing, colloquially known as a "swing wing", is an airplane wing, or set of wings, that may be modified during flight, swept back and then
Variable-sweep_wing
Statistical modeling method
(dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable is a simple
Linear_regression
Statistical test for causality
computed from S {\displaystyle S} random realizations (surrogates) of X {\displaystyle X} . CCM is used to detect if two variables belong to the same dynamical
Convergent_cross_mapping
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
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 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
Type of gradient noise in computer graphics
limited to: procedurally generating terrain, applying pseudo-random changes to a variable, and assisting in the creation of image textures. It is most
Perlin_noise
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
Effect of variables' uncertainties on the uncertainty of a function based on them
uncertainty is the effect of variables' uncertainties on the uncertainty of a function based on them. When the variables are the values of experimental
Propagation_of_uncertainty
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
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
Branch of mathematics
probability theory to determine the expectation value of a continuous random variable given a probability density function. In analytic geometry, the study
Calculus
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
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
Mathematical methods used in Bayesian inference and machine learning
used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various
Variational_Bayesian_methods
Stochastic process generalizing Brownian motion
0 ≤ s1 < t1 ≤ s2 < t2 then Wt1 − Ws1 and Wt2 − Ws2 are independent random variables, and the similar condition holds for n increments. Condition 2 can
Wiener_process
Type of simulation
happen without any delay. Otherwise, the state variable teller-status is set to "available". The random variables that need to be characterized to model this
Discrete-event_simulation
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
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
"Pushed forward" from one measurable space to another
codomain is 1. This means that random variables can be composed ad infinitum and they will always remain random variables and endow the codomain spaces
Pushforward_measure
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
Male
English
 Variant spelling of Middle English Randulf, RANDOLF means "shield-wolf." Compare with other forms of Randolf.
Male
Norwegian
 Norwegian form of Old Norse Arnþórr, ANDOR means "eagle of Thor." Compare with another form of Andor.
Surname or Lastname
English
English : patronymic from Rand 1.
Male
English
Medieval form of English Randolf, RANDAL means "shield-wolf."
Boy/Male
English
Son of Rand.
Surname or Lastname
English
English : variant of Ransom.
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : patronymic from the Middle English personal name Rand(e) (see Rand 1).
Male
English
Pet form of English Randall and Randolph, both RANDY means "shield-wolf." Compare with feminine Randy.
Surname or Lastname
English
English : variant spelling of Randall.Americanized spelling of Randel.
Male
Scandinavian
 Scandinavian form of Old Norse Randolfr, RANDOLF means "shield-wolf." Compare with another form of Randolf.
Surname or Lastname
English
English : probably a variant of Crandon, a habitational name from Crandon in Somerset or Crandean in Falmer, Sussex. Compare Grandin.
Surname or Lastname
English
English : unexplained; perhaps a variant of Francom.
Male
Hungarian
 Variant spelling of Hungarian András, ANDOR means "man; warrior." Compare with another form of Andor.
Surname or Lastname
English
English : variant of Rand 1, from the Old French oblique case.
Female
English
Short form of English Miranda, RANDA means "worthy of admiration."Â
Surname or Lastname
English
English : variant of Brandon.
Female
English
Pet form of English Miranda, RANDY means "worthy of admiration."Â Compare with masculine Randy.Â
Female
English
Variant spelling of English Randy, RANDI means "worthy of admiration."
Boy/Male
English American
Son of Rand.
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
Girl/Female
Indian, Sanskrit
Like Snow
Girl/Female
Indian
Flame, Lamp
Girl/Female
Hindu
Goddess Saraswati
Girl/Female
British, English
Wealthy Defender
Boy/Male
Tamil
Lord Shiva, Sentiment
Girl/Female
Biblical
Governor of a fourth part.
Boy/Male
Arabic, Bengali, Gujarati, Hindu, Indian
King of Swan; Ray of Sunlight
Girl/Female
Indian
Beautiful
Boy/Male
Australian, Hebrew, Jewish
Deer
Girl/Female
Hindu
Ascending, Growing
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
COMPLEX RANDOM-VARIABLE
a.
Complex, complicated.
imp. & p. p.
of Compile
n.
To redeem from captivity, servitude, punishment, or forfeit, by paying a price; to buy out of servitude or penalty; to rescue; to deliver; as, to ransom prisoners from an enemy.
a.
Not complex; uncompounded; simple.
v. i.
To go or stray at random.
imp. & p. p.
of Couple
a.
Going at random or by chance; done or made at hazard, or without settled direction, aim, or purpose; hazarded without previous calculation; left to chance; haphazard; as, a random guess.
n.
Random.
n.
Distance to which a missile is cast; range; reach; as, the random of a rifle ball.
n.
A roving motion; course without definite direction; want of direction, rule, or method; hazard; chance; -- commonly used in the phrase at random, that is, without a settled point of direction; at hazard.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
adv.
In a complex manner; not simply.
n.
To exact a ransom for, or a payment on.
imp. & p. p.
of Comply
adv.
In a random manner.
n.
The release of a captive, or of captured property, by payment of a consideration; redemption; as, prisoners hopeless of ransom.
n.
A complex; an aggregate of parts; a complication.
a.
Repeatedly compound; made up of complex constituents.
a.
Intricate; entangled; complicated; complex.