# NoncentralStudentTDistribution

represents a noncentral Student distribution with ν degrees of freedom and noncentrality parameter δ.

# Background & Context

• represents a continuous statistical distribution defined and supported over the set of real numbers and parametrized by a real number δ (called a "noncentrality parameter") and by a positive real number ν (the "degrees of freedom") that together determine the overall behavior of its probability density function (PDF). In general, the PDF of a noncentral Student -distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of ν and δ. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) NoncentralStudentTDistribution is a perhaps-skewed generalization of the Student -distribution (StudentTDistribution, sometimes referred to as the centralized Student -distribution) and is sometimes referred to as the noncentral -distribution.
• The noncentral -distribution was first devised in the 1930s, though its first thorough quantitative analysis came in a 1940 article by Johnson and Welch. Originally, the distribution was realized as the distribution describing the behavior of a random variate of the form where , where wChiSquareDistribution[ν]/ν, and where δ is the noncentrality parameter. In the years since, the noncentral -distribution has found a number of fundamental uses throughout statistics such as Bayesian inference, normality testing, and modeling the power of null hypothesis test for a population chosen according to a (central) -distribution. The distribution is also used in quality control, microbiology, macroeconomics, and pharmaceuticals.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a noncentral -distribution. Distributed[x,NoncentralStudentTDistribution[ν,δ]], written more concisely as xNoncentralStudentTDistribution[ν,δ], can be used to assert that a random variable x is distributed according to a noncentral -distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for noncentral -distributions may be given using PDF[NoncentralStudentTDistribution[ν,δ],x] and CDF[NoncentralStudentTDistribution[ν,δ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a noncentral -distribution, EstimatedDistribution to estimate a noncentral parametric distribution from given data, and FindDistributionParameters to fit data to a noncentral -distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic noncentral -distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic noncentral -distribution.
• TransformedDistribution can be used to represent a transformed noncentral -distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a noncentral -distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving noncentral -distributions.
• NoncentralStudentTDistribution is related to a number of other distributions. It is an immediate generalization of StudentTDistribution in the sense that the PDF of is precisely that of . NoncentralStudentTDistribution is also related to NormalDistribution, ChiSquareDistribution, PearsonDistribution, MultivariateTDistribution, FRatioDistribution, GammaDistribution, and HotellingTSquareDistribution.

# Examples

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## Basic Examples(3)

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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