NationMaster - Encyclopedia: Student's t distribution

In probability and statistics, the t-distribution or Student's t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. It is the basis of the popular Student's t-tests for the statistical significance of the difference between two sample means, and for confidence intervals for the difference between two population means. The Student's t-distribution is a special case of the generalised hyperbolic distribution. ImageMetadata File history File links Download high resolution version (918x567, 47 KB) Summary eprsonal production Free use and distribution Licensing Cleanup This image needs to be cleaned up to conform to a higher standard of quality. ... Image File history File links Download high resolution version (1200x1200, 18 KB) Summary The CDF of the t-distribution bitmap(file=t_distributionCDF.png,type=png256,width=4,height=4,res=300,pointsize=12) par(mar=c(3,3,1,1)) x <- seq(-5,5,len=1000) plot(range(x),c... Please refer to Real vs. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ... In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Probability is the likelihood that something is the case or will happen. ... A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... Sample size, usually designated N, is the number of repeated measurements in a statistical sample. ... A t test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ... In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In this diagram, the bars represent observation means and the red lines represent the confidence intervals surrounding them. ... The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ...

The derivation of the t-distribution was first published in 1908 by William Sealy Gosset, while he worked at a Guinness Brewery in Dublin. He was not allowed to publish under his own name, so the paper was written under the pseudonym Student. The t-test and the associated theory became well-known through the work of R.A. Fisher, who called the distribution "Student's distribution". William Sealy Gosset, 1876-1937 William Sealy Gosset (June 13, 1876â€“October 16, 1937) was an English chemist and statistician, best known by his pen name Student and for his work on Students t-distribution. ... St. ... Dublin city centre at night WGS-84 (GPS) Coordinates: , Statistics Province: Leinster County: DÃ¡il Ã‰ireann: Dublin Central, Dublin North Central, Dublin North East, Dublin North West, Dublin South Central, Dublin South East European Parliament: Dublin Dialling Code: +353 1 Postal District(s): D1-24, D6W Area: 114. ... A t test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ... Sir Ronald Fisher Sir Ronald Aylmer Fisher, FRS (February 17, 1890–July 29, 1962) was an extraordinarily talented evolutionary biologist, geneticist and statistician. ...

Student's distribution arises when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data. Textbook problems treating the standard deviation as if it were known are of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

Why Student's t-distribution

Confidence intervals and hypothesis tests rely on Student's t-distribution to cope with uncertainty resulting from estimating the standard deviation from a sample, whereas if the population standard deviation were known, a normal distribution would be used. In this diagram, the bars represent observation means and the red lines represent the confidence intervals surrounding them. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

How Student's t-distribution comes about

Suppose X₁, ..., X_n are independent random variables that are normally distributed with expected value μ and variance σ². Let In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

is normally distributed with mean 0 and variance 1, since the sample mean $scriptstyle overline{X}_n$ is normally distributed with mean

μ

and standard deviation $scriptstylesigma/sqrt{n}$ .

While similar to Z, the variance $scriptstyle S_n^2$ is estimated. Thus $scriptstyle S_n^2$ has a $scriptstylechi_n^2$ distribution (which introduces uncertainty into the denominator). Gosset's work showed that T has the probability density function In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

The distribution of T is now called the t-distribution. The parameter ν is called the number of degrees of freedom. The distribution depends on ν, but not μ or σ; the lack of dependence on μ and σ is what makes the t-distribution important in both theory and practice. Γ is the Gamma function. This article or section is in need of attention from an expert on the subject. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...

It should be noted that the term for 0 < k < ν, k even, may be simplified using the properties of the Gamma function to The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...

Confidence intervals derived from Student's t-distribution

when T has a t-distribution with n − 1 degrees of freedom. This is the same as

so A is the "95th percentile" of this probability distribution, or

A = t (0.05, n - 1)

. Then

is a 90-percent confidence interval for μ. Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use the t-distribution to examine whether the confidence limits on that mean include some theoretically predicted value - such as the value predicted on a null hypothesis. In this diagram, the bars represent observation means and the red lines represent the confidence intervals surrounding them. ... In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ...

It is this result that is used in the Student's t-tests: since the difference between the means of samples from two normal distributions is itself distributed normally, the t-distribution can be used to examine whether that difference can reasonably be supposed to be zero. A t test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ...

If the data are normally distributed, the one-sided (1 − a)-upper confidence limit (UCL) of the mean, can be calculated using the following equation:

The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, $overline{X}_n$ being the mean of the set of observations, the probability that the mean of the distribution is inferior to

UCL 1 - a

is equal to the confidence level

1 - a .

A number of other statistics can be shown to have t-distributions for samples of moderate size under null hypotheses that are of interest, so that the t-distribution forms the basis for significance tests in other situations as well as when examining the differences between means. For example, the distribution of Spearman's rank correlation coefficient, rho, in the null case (zero correlation) is well approximated by the t distribution for sample sizes above about 20. In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... In statistics, Spearmans rank correlation coefficient, named after Charles Spearman and often denoted by the Greek letter Ï (rho), is a non-parametric measure of correlation â€“ that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about the frequency...

See prediction interval for another example of the use of this distribution. In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter. ...

Student's distribution probability function and p-value

The distribution

A (t | ν)

is used when testing whether a

t

-statistic that measures the difference between two means is statistically significant. This is used in a variety of situations, particularly in t-tests. For the statistic t, with

ν

degrees of freedom,

A (t | ν)

is the probability that t would be less than the observed value if the two means were the same. It is given by the following formula: A t-test is any statistical hypothesis test in which the test statistic has a Students t-distribution if the null hypothesis is true. ...

where B is the Beta function. There is a relation to the incomplete beta function

I x (a, b)

as follows: In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ... In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ...

The exact p-value is given by In statistical hypothesis testing, the p-value of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ...

Further theory

Gosset's result can be stated more generally. (See, for example, Hogg and Craig, Sections 4.4 and 4.8.) Let Z have a normal distribution with mean 0 and variance 1. Let V have a chi-square distribution with ν degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the chi-square distribution (also chi-squared or Ï‡2 distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ... In statistics, Cochrans theorem is used in the analysis of variance. ...

For a t-distribution with ν degrees of freedom, the expected value is 0, and its variance is ν/(ν − 2) if ν > 2. The skewness is 0 and the kurtosis is 6/(ν − 4) if ν > 4. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ...

The cumulative distribution function is given by an incomplete beta function, In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. ...

The t-distribution is related to the F-distribution as follows: the square of a value of t with ν degrees of freedom is distributed as F with 1 and ν degrees of freedom. In statistics and probability, the F-distribution is a continuous probability distribution. ...

The overall shape of the probability density function of the t-distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean 0 and variance 1. The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

The following images show the density of the t-distribution for increasing values of ν. The normal distribution is shown as a blue line for comparison.; Note that the t-distribution (red line) becomes closer to the normal distribution as ν increases. For ν = 30 the t-distribution is almost the same as the normal distribution.

plot of t-distribution, 1 df File links The following pages link to this file: Students t-distribution Categories: GFDL images | Probability distributions images ... plot of t-distribution, 2 df File links The following pages link to this file: Students t-distribution Categories: GFDL images | Probability distributions images ... plot of t-distribution, 3 df File links The following pages link to this file: Students t-distribution Categories: GFDL images | Probability distributions images ... plot of t-distribution, 5 df File links The following pages link to this file: Students t-distribution Categories: GFDL images | Probability distributions images ... plot of t-distribution, 10 df File links The following pages link to this file: Students t-distribution Categories: GFDL images | Probability distributions images ... plot of t-distribution, 30 df File links The following pages link to this file: Students t-distribution Categories: GFDL images | Probability distributions images ...

Table of selected values

The following table lists a few selected values for t-distributions with ν degrees of freedom for the 90%, 95%, 97.5%, and 99.5% one-sided confidence intervals. For an example of how to read this table, take the fourth row, which begins with 4; that means ν, the number of degrees of freedom, is 4 (and if we are dealing, as above, with n values with a fixed sum, n = 5). Take the fourth entry, in the column headed 90%. The value of that entry is "1.533". Then the probability that T is less than 1.533 is 90% or Pr(-∞ <T < 1.533) = 0.9; the entry does not mean (as it might with other distributions) Pr(−1.533 < T < 1.533) = 0.9.

Note that the last row also gives critical points: a t-distribution with infinitely-many degrees of freedom is a normal distribution. (See below: Related distributions).

The number at the beginning of each row in the table above is ν which has been defined above as n − 1. The percentage along the top is 100%(1 − α). The numbers in the main body of the table are t_α,ν. If a quantity T is distributed as a Student's t distribution with ν degrees of freedom, then there is a probability 1 − α that T will be less than t_α,ν.(Calculated as for a one-tailed or one-sided test as opposed to a two-tailed test.) The two-tailed test is the test of a given statistical hypothesis in which a value of the statistic that is either sufficiently small or sufficiently large will lead to rejection of the hypothesis tested. ...

For example, given a sample with a sample variance 2 and sample mean of 10, taken from a sample set of 11 (10 degrees of freedom), using the formula

(In other words, on average, 90% of the times that an upper threshold is calculated by this method, the true mean lies below this upper threshold.) And, still at 90% confidence, we have a true mean lying over

(In other words, on average, 90% of the times that a lower threshold is calculated by this method, the true mean lies above this lower threshold.) So that at 80% confidence, we have a true mean lying between

(In other words, on average, 80% of the times that upper and lower thresholds are calculated by this method, the true mean is both below the upper threshold and above the lower threshold. This is not the same thing as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method -- see confidence interval and prosecutor's fallacy.) In this diagram, the bars represent observation means and the red lines represent the confidence intervals surrounding them. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

For information on the inverse cumulative distribution function see Quantile function. This article or section does not cite any references or sources. ...

Special cases

ν = 1

See Cauchy distribution The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...

ν = 2

Robust parametric modelling

The t-distribution is often used as an alternative to the normal distribution as a model for data. It is frequently the case that real data have heavier tails than the normal distribution allows for. The classical approach was to identify outliers and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the t-distribution is a natural choice of model for such data and provides a parametric approach to robust statistics. Robust statistics provides an alternative approach to classical statistical methods. ...

Lange et al explored the use of the t-distribution for robust modelling of heavy tailed data in a variety of contexts. A Bayesian account can be found in Gelman et al. The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Some authors report that values between 3 and 9 are often good choices. Venables and Ripley suggest that a value of 5 is often a good choice.

Related distributions

The scale-inverse-chi-square distribution arises in Bayesian statistics (spam filtering in particular). ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...

See also

A t test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... In statistics, Hotellings T-square statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ... Where the hell is this ? ... In statistics, a multivariate Student distribution is a multivariate generalization of the Students t-distribution. ... In this diagram, the bars represent observation means and the red lines represent the confidence intervals surrounding them. ...

References

William Sealy Gosset, 1876-1937 William Sealy Gosset (June 13, 1876â€“October 16, 1937) was an English chemist and statistician, best known by his pen name Student and for his work on Students t-distribution. ... Biometrika is a scientific journal established in 1901 by Francis Galton, Karl Pearson and W. F. R. Weldon to promote the study of biometrics, the statistical analysis of hereditary phenomena. ... Sir Ronald Aylmer Fisher, FRS (17 February 1890 â€“ 29 July 1962) was a British statistician, evolutionary biologist, and geneticist. ... Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ...

External links

Categories: Continuous distributions | Special functions Image File history File links Bvn-small. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... Univariate describes a concept in statistics or econometrics. ... A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Î©, P). ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A logarithmic scale bar. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of... Often confused with the multinomial distribution. ... In probability theory, a compound Poisson distribution is the probability distribution of a Poisson-distibuted number of independent identically-distributed random variables. ... The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... // In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ... In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ... The Skellam distribution is the discrete probability distribution of the difference N1 âˆ’ N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values Î¼1 and Î¼2. ... In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ... In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... The multivariate Polya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... 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The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... A phase-type distribution is a probability distribution that results from a system one or more inter-related poisson processes occurring in sequence, or phases. ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... 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In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, FrÃ©chet and Weibull families also known as type I, II and III extreme value distributions. ... The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ... In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ... In probability theory and statistics, the half-logistic distribution is a continuous probability distributionâ€”the distribution of the absolute value of a random variable following the logistic distribution. ... In statistics, Hotellings T-square statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ... In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable X is given by: Where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution... The hypoexponential distribution is a generalization of Erlang distribution in the sense that the n exponential distributions may have different rates. ... 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The probability distribution for Landau random variates is defined analytically by the complex integral, For numerical purposes it is more convenient to use the following equivalent form of the integral, From GSL manual, used under GFDL. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory and statistics, the LÃ©vy distribution, named after Paul Pierre LÃ©vy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ... In probability theory, a LÃ©vy skew alpha-stable distribution or just stable distribution, developed by Paul LÃ©vy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... There are very few or no other articles that link to this one. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. ... The normal-inverse Gaussian distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... This article or section is incomplete and may require expansion and/or cleanup. ... A phase-type distribution is a probability distribution that results from a system one or more inter-related poisson processes occurring in sequence, or phases. ... In probability theory, the polar distribution is the probability distribution of angles occurring in a set of two-dimensional vectors, denoted by It is usually graphically represented as a closed curve , where the radius equals the probability . ... In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... NB: The information in this article should be reviewed. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and respectively. ... In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ... In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). ... In probability theory, the Type-1 Gumbel distribution function is for . Reference Taken from the gsl-ref_19. ... In probability theory, the Type-2 Gumbel distribution function is for . Based on gsl-ref_19. ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... The variance-gamma distribution is continuous probability distribution that is defined as the normal varianc

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