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Probability distribution
In probability theory and statistics, a normal variance-mean mixture with mixing probability density g {\displaystyle g} is the continuous probability
Normal_variance-mean_mixture
Continuous probability distribution
a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution
Normal-inverse Gaussian distribution
Normal-inverse_Gaussian_distribution
Name for several different families of probability distributions
\alpha )} . Logistic-beta distribution admits the following normal variance-mean mixture representation: f ( x ; α , β ) = 1 B ( α , β ) e − β x ( 1 +
Generalized logistic distribution
Generalized_logistic_distribution
Probability distribution
of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples
Normal_distribution
Continuous probability distribution
normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution
Variance-gamma_distribution
Probability distribution
law). The log-normal distribution is the maximum entropy probability distribution for a random variate X—for which the mean and variance of ln X are specified
Log-normal_distribution
Continuous probability distribution
(GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian
Generalised hyperbolic distribution
Generalised_hyperbolic_distribution
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
moment, μ k / σ k {\displaystyle \mu _{k}/\sigma ^{k}} Variance-to-mean ratio (or relative variance), σ 2 / μ {\displaystyle \sigma ^{2}/\mu } Fano factor
Coefficient_of_variation
Generalization of gamma distribution to multiple dimensions
off-diagonal element is less familiar but can be identified as a normal variance-mean mixture where the mixing density is a χ2 distribution. The corresponding
Wishart_distribution
Generalization of the one-dimensional normal distribution to higher dimensions
{X} } has a univariate normal distribution, where a univariate normal distribution with zero variance is a point mass on its mean. There is a k-vector μ
Multivariate normal distribution
Multivariate_normal_distribution
Theorem in probability theory
total variance is a fundamental result in probability theory that expresses the variance of a random variable Y in terms of its conditional variances and
Law_of_total_variance
Normal curve equivalent Normal distribution Normal probability plot – see also rankit Normal score – see also rankit and Z score Normal variance-mean
List_of_statistics_articles
Statistical concept
components are Gaussian distributions, there will be a mean and variance for each component. If the mixture components are categorical distributions (e.g., when
Mixture_model
Probability distribution
}{\alpha +\beta }}\right)} converges in distribution to a normal distribution with mean 0 and variance α β ( α + β ) 3 {\displaystyle {\tfrac {\alpha \beta
Beta_distribution
Statistical hypothesis test
population mean. The assumptions underlying a t-test in the simplest form above are that: X follows a normal distribution with mean μ and variance σ2/n. s2(n − 1)/σ2
Student's_t-test
Concept in probability theory
prior for unknown mean and variance, but with a fixed, linear relationship between them, is found in the normal variance-mean mixture, with the generalized
Conjugate_prior
Type of probability distribution
the mean of the i-th component. In the case of a mixture of one-dimensional distributions with weights wi, means μi and variances σi2, the total mean and
Mixture_distribution
Concept in statistics
by utilizing the EM-algorithm. Gaussian scale mixtures: Compounding a normal distribution with variance distributed according to an inverse gamma distribution
Compound probability distribution
Compound_probability_distribution
Fourth standardized moment in statistics
normality. For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see variance. Pearson's definition
Kurtosis
Probability distribution with more than one mode
parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two normal distributions with equal standard deviations
Multimodal_distribution
Middle quantile of a data set or probability distribution
minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean. A median
Median
Probability distribution on the circle
} The parameters μ and 1/κ are analogous to μ and σ2 (the mean and variance) in the normal distribution: μ is a measure of location (the distribution
Von_Mises_distribution
Inverse of the average of the inverses of a set of numbers
harmonic mean is calculated as above. Both the mean and the variance may be infinite (if it includes at least one term of the form 1/0). The mean of the
Harmonic_mean
Mixture of discrete and continuous distributions
the cdf of a normal distribution with mean μ {\displaystyle \mu } and variance σ 2 {\displaystyle \sigma ^{2}} . The mean and variance of the rectified
Rectified Gaussian distribution
Rectified_Gaussian_distribution
Presence of greater variability in a data set than would be expected
normal distribution with the exact variance – the normal distribution is a two-parameter model, with mean and variance. Thus, in the absence of an underlying
Overdispersion
Family of continuous probability distributions
distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance-mean mixture. Let the prior distribution
Generalized inverse Gaussian distribution
Generalized_inverse_Gaussian_distribution
Probability distribution
converges to normal distribution with mean μ = αθ and variance σ2 = αθ2. The gamma distribution is the conjugate prior for the precision of the normal distribution
Gamma_distribution
Vector quantization algorithm minimizing the sum of squared deviations
within-cluster variances (squared Euclidean distances), but not regular Euclidean distances, which would be the more difficult Weber problem: the mean optimizes
K-means_clustering
distributed variables with finite mean and variance is approximately normal. The normal-exponential-gamma distribution The normal-inverse Gaussian distribution
List of probability distributions
List_of_probability_distributions
Measure of distance to normality
normality. Out of all probability distributions with a given mean and variance, the Gaussian or normal distribution is the one with the highest entropy.[clarification
Negentropy
Statistical distance measure
{\displaystyle P} moves away from the mean along each principal component axis. If each of these axes is re-scaled to have unit variance, and whitened to be uncorrelated
Mahalanobis_distance
Subdiscipline of statistics
root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution
Directional_statistics
Variation in the time intervals between heartbeats
of heart rate or the heart period (R–R interval) partitions the total variance (the "power") of a continuous series of beats into its frequency components
Heart_rate_variability
Mathematical function for the probability a given outcome occurs in an experiment
( X < x ) = q {\displaystyle P(X<x)=q} . Variance: the second moment of the random variable about its mean; an important measure of the dispersion of
Probability_distribution
Statistical model allowing for frequent zero values
is the probability of extra zeros. The mean is ( 1 − π ) λ {\displaystyle (1-\pi )\lambda } and the variance is λ ( 1 − π ) ( 1 + π λ ) {\displaystyle
Zero-inflated_model
British eugenist, polymath, and behavioural geneticist (1822–1911)
tendency, or mean, and a spread around this central value, or variance. In the late 1860s, Galton conceived of a measure to quantify normal variation: the
Francis_Galton
Iterative method for finding maximum likelihood estimates in statistical models
solutions that may be found by EM in a mixture model involves setting one of the components to have zero variance and the mean parameter for the same component
Expectation–maximization algorithm
Expectation–maximization_algorithm
Type of signal in signal processing
sequence of serially uncorrelated random variables with a mean of zero and a finite variance; a single realization of white noise is a random shock. In
White_noise
Quality measure of a statistical method
practice, efficient estimators exist for: the mean μ of the normal distribution (but not the variance σ2), parameter λ of the Poisson distribution, the
Efficiency_(statistics)
Josephson et al. limited themselves to considering two normal mixtures with the same component variances and mixing proportions. As a consequence, their proposal
Sexual_dimorphism_measures
Type of probability distribution
constant C {\displaystyle C} such that given any number of independent mean-zero variance-one subgaussian random variables X 1 , … , X N {\displaystyle X_{1}
Sub-Gaussian_distribution
Parameter estimation technique in statistics
(}X_{i}-{\overline {X}}{\big )}^{2},\end{aligned}}} which are the sample mean and the (biased) sample variance. In this case, they coincide with the maximum likelihood
Method of moments (statistics)
Method_of_moments_(statistics)
Mathematical concept
distributed around the mean while the frequency of occurrence of data farther away from the mean diminishes, one may for example select the normal distribution
Probability distribution fitting
Probability_distribution_fitting
N-th root of the product of n numbers
Arithmetic-geometric mean Generalized mean Geometric mean theorem Geometric standard deviation Harmonic mean Heronian mean Heteroscedasticity Log-normal distribution
Geometric_mean
Aspect of probability theory
distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the
Sum of normally distributed random variables
Sum_of_normally_distributed_random_variables
Statistical function that defines the quantiles of a probability distribution
Two four-parametric quantile mixtures, the normal-polynomial quantile mixture and the Cauchy-polynomial quantile mixture, are presented by Karvanen. The
Quantile_function
Probability distribution
(50% efficient). These mean and variance parameter estimates, together with parallel estimates for X, can be applied to Normal or Binomial approximations
Ratio_distribution
In computer vision and image processing
method models the histogram of the image as a mixture of two normal distributions with equal variance and equal size. However, Otsu's thresholding may
Otsu's_method
Probability distribution
sample variance exceeds the sample mean. In such cases, the observations are overdispersed with respect to a Poisson distribution, for which the mean is equal
Negative binomial distribution
Negative_binomial_distribution
Probability distribution that has the most entropy of a class
upper bound. Start with a normal distribution of the specified mean and variance. To introduce a positive skew, perturb the normal distribution upward by
Maximum entropy probability distribution
Maximum_entropy_probability_distribution
Probability distribution
each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized
Rayleigh_distribution
Correlation of a signal with a time-shifted copy of itself, as a function of shift
may not be well defined. Suppose that the process has mean μ t {\displaystyle \mu _{t}} and variance σ t 2 {\displaystyle \sigma _{t}^{2}} at time t {\displaystyle
Autocorrelation
Observation far apart from others in statistics and data science
this is modeled by a mixture model. In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable
Outlier
Specialized form of regression analysis, in statistics
observations are from a specified normal distribution, but a small proportion are from a normal distribution with much higher variance. That is, residuals have
Robust_regression
Concept in statistics
the mean. When estimating a scale parameter, using a trimmed estimator as a robust measures of scale, such as to estimate the population variance or population
Trimmed_estimator
Family of probability distributions related to the normal distribution
the family of normal distributions includes the standard normal distribution N(0, 1) with mean 0 and variance 1, as well as other normal distributions
Exponential_family
Set of quantities in probability theory
cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central
Cumulant
Study of the inheritance of continuously variable traits
employ many other statistical methods (such as the effect size, the mean and the variance) to link phenotypes (attributes) to genotypes. Some phenotypes may
Quantitative_genetics
Statistical measure
follow a normal distribution. Central tendency Invariant estimator Location parameter Location-scale family Mean-preserving spread Scale mixture Shape parameter
Scale_parameter
Method to find best fit of a time-series model
be independent of each other and constant in mean and variance over time. (Plotting the mean and variance of residuals over time and performing a Ljung–Box
Box–Jenkins_method
Distribution of an uncertain quantity
entropy prior given that the density is normalized with mean zero and unit variance is the standard normal distribution. The principle of minimum cross-entropy
Prior_probability
Notion in statistics
unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. The role
Fisher_information
Statistical model for count data
the variance is equal to the mean made by the Poisson model. The traditional negative binomial regression model is based on the Poisson-gamma mixture distribution
Poisson_regression
Type of Monte Carlo algorithms for signal processing and statistical inference
uneven. Several adaptive resampling criteria can be used including the variance of the weights and the relative entropy concerning the uniform distribution
Particle_filter
Distribution of variables which satisfies a stability property under linear combinations
\rightarrow 0} . The distributions have undefined variance for α < 2 {\displaystyle \alpha <2} , and undefined mean for α ≤ 1 {\displaystyle \alpha \leq 1} .
Stable_distribution
Form of causal modeling that fit networks of constructs to data
of methodologies that seeks to represent hypotheses about the means, variances, and covariances of observed data in terms of a smaller number of 'structural'
Structural_equation_modeling
Set of probability distributions
with θ = ( A ′ ) − 1 ( μ ) {\displaystyle \theta =(A')^{-1}(\mu )} Mean and variance of Y ∼ E D ( μ , σ 2 ) {\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma
Exponential_dispersion_model
Variable representing a random phenomenon
makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments
Random_variable
Vector with non-negative entries that add up to one
{\displaystyle 0} . The bounds on variance show that as the number of possible outcomes n {\displaystyle n} increases, the variance necessarily decreases toward
Probability_vector
Transforming data by taking the logarithm
population mean is to take the sample mean plus or minus two standard error units. However, the constant factor 2 used here is particular to the normal distribution
Log transformation (statistics)
Log_transformation_(statistics)
Empirical law on the variance of species in a habitat
ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship
Taylor's_law
Probability distribution
approaches a log-normal distribution. Although the log-normal distribution has finite moments, for any finite degrees of freedom, the mean and variance and all
Log-t_distribution
Family of distributions that generalize the multivariate normal distribution
finite-variance assumptions, an extension of Cochran's theorem (on the distribution of quadratic forms) holds. An elliptical distribution with a zero mean and
Elliptical_distribution
Time series model
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
Autoregressive conditional heteroskedasticity
Autoregressive_conditional_heteroskedasticity
Least squares approximation of linear functions to data
{\displaystyle \varepsilon } , are uncorrelated, have a mean of zero and a constant variance, σ {\displaystyle \sigma } , the Gauss–Markov theorem states
Linear_least_squares
Probability distribution
mean and variance of the inverse-gamma distribution, Inv-Gamma ( n , λ ) {\textstyle {\mbox{Inv-Gamma}}(n,\lambda )} . The uniformly minimum-variance
Exponential_distribution
Probabilistic classification algorithm
class, and then the mean and variance of x {\displaystyle x} is computed in each class. Let μ k {\displaystyle \mu _{k}} be the mean of the values in x
Naive_Bayes_classifier
Concept in probability theory
normal variate divided by an independent standard uniform variate. In other words, if the random variable Z has a normal distribution with zero mean and
Slash_distribution
Concept in statistics
axis). The grey curve is the true density (a normal density with mean 0 and variance 1). In comparison, the red curve is undersmoothed since it contains
Kernel_density_estimation
Method of statistical inference
completely. Example: A hypothesis specifying a normal distribution with a specified mean and an unspecified variance. The simple/composite distinction was made
Statistical_hypothesis_test
Observation that in many real-life datasets, the leading digit is likely to be small
log-normal distribution depends on the mean and the variance of the distribution. The variance has a much greater effect on the fit than does the mean. Larger
Benford's_law
increases X n {\displaystyle X_{n}} are normal IID variables, their sum will be normally distributed with mean nμ and variance nσ2. The probability that the crack
Birnbaum–Saunders distribution
Birnbaum–Saunders_distribution
Statistical distribution for dependence between random variables
S2CID 154138333. Low, R.K.Y.; Faff, R.; Aas, K. (2016). "Enhancing mean–variance portfolio selection by modeling distributional asymmetries" (PDF). Journal
Copula_(statistics)
Signal processing computational method
x {\displaystyle \mathbf {x} } into a new mixture z {\displaystyle \mathbf {z} } , which has unit variance, and z = ( z 1 , z 2 , … , z M ) T {\displaystyle
Independent component analysis
Independent_component_analysis
Experimental design that is optimal with respect to some statistical criterion
parameters, however, the mean of the parameter-estimator is a vector and its variance is a matrix. The inverse matrix of the variance-matrix is called the
Optimal_experimental_design
Statistical model
{\displaystyle K(\theta ,x^{*},x^{*})} is the variance at point x* as dictated by θ. Practically, the posterior mean estimate of f ( x ∗ ) {\displaystyle f(x^{*})}
Gaussian_process
Ballistics measure of a weapon system's precision
the square root of the mean square error (MSE). The MSE will be the sum of the variance of the range error plus the variance of the azimuth error plus
Circular_error_probable
Exact statistical hypothesis test
100 % {\displaystyle \alpha \times 100\%} significance level. To exploit variance reduction with paired samples, a paired permutation test must be applied
Permutation_test
Method of estimating the parameters of a statistical model
the MAP is the optimal point estimator. In many types of models, such as mixture models, the posterior may be multi-modal. In such a case, the usual recommendation
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Function related to statistics and probability theory
(\theta )\right|\,} is finite. This ensures that the score has a finite variance. The above conditions are sufficient, but not necessary. That is, a model
Likelihood_function
Mathematical methods used in Bayesian inference and machine learning
For example, a typical Gaussian mixture model will have parameters for the mean and variance of each of the mixture components. EM would directly estimate
Variational_Bayesian_methods
Method of estimating a statistical model's parameters
S_{n}({\hat {\theta }})=M_{n}({\hat {\theta }})} has the same asymptotic mean and variance as in the known case. However, the test statistic to be used requires
Maximum_spacing_estimation
Type of statistics
parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model
Robust_statistics
Grouping a set of objects by similarity
cluster by a single mean vector. Distribution models: clusters are modeled using statistical distributions, such as multivariate normal distributions used
Cluster_analysis
Statistical method for handling multiple comparisons
defined as: Q ′ = E [ V ] R {\displaystyle Q'={\frac {E[V]}{R}}} . This is a mixture of expectations and realizations, and has the problem of control for m
False_discovery_rate
Branch of mathematics concerning probability
X_{1},X_{2},\dots \,} be independent random variables with mean μ {\displaystyle \mu } and variance σ 2 > 0. {\displaystyle \sigma ^{2}>0.\,} Then the sequence
Probability_theory
Frequency with which an engineered system or component fails
"mean time between critical failures" (MTBCF). Combining failure or hazard rates that are time-dependent is more complicated. For example, mixtures of
Failure_rate
Class of statistical estimators
iterative reweighting schemes, and Simon Newcomb (1886) experimented with mixtures of distributions for regression. By the late 19th century, Smith (1888)
M-estimator
Function for integral Fourier-like transform
{\displaystyle \sigma _{1}^{2}} is the variance of "significant" coefficients and σ 2 2 {\displaystyle \sigma _{2}^{2}} is the variance of "insignificant" coefficients
Wavelet
Statistical indicators of the deviation of a sample
distribution without outliers, such as a normal distribution. However, they have superior efficiency for data drawn from a mixture distribution or from a heavy-tailed
Robust_measures_of_scale
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