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MATHEMATICA STUDENT PDF

CopulaDistribution can be used to build higher-dimensional distributions that contain a Student distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Student distributions.

TransformedDistribution can be used to represent a transformed Student 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.

ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Student distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Student distribution.

DistributionFitTest can be used to test if a given dataset is consistent with a Student distribution, EstimatedDistribution to estimate a Student parametric distribution from given data, and FindDistributionParameters to fit data to a Student distribution.

The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.

The probability density and cumulative distribution functions for Student distributions may be given using PDF, x ] and CDF, x ].

Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation. Distributed ], written more concisely as x StudentTDistribution, can be used to assert that a random variable x is distributed according to a Student distribution.

RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Student distribution.

The distribution has also found extensive use across a number of different fields to model phenomena including stock price fluctuations, phase derivatives of telecommunications components, noise models, and image analysis. The -distribution is widely used throughout statistics and is an often-utilized tool in hypothesis testing, analysis of variance tests, Bayesian analysis, and stochastic processes. Gosset showed that for integer ν, the Student distribution is precisely the distribution of the deviation of the observed mean from the true population mean given a sample of ν normalized and normally-distributed random variates.

The Student distribution was first devised by English statistician William Gosset (published under the pseudonym "Student") in 1908.

(This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The one-parameter form StudentTDistribution is equivalent to StudentTDistribution and is sometimes referred to as "the" Student distribution, while the three-parameter form StudentTDistribution is sometimes referred to as the generalized Student distribution.

MATHEMATICA STUDENT PDF

In addition, the tails of the PDF are "fat", in the sense that the PDF decreases algebraically rather than decreasing exponentially for large values of. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of μ, σ, and ν.

In general, the PDF of a Student distribution is unimodal with a single "peak" (i.e. StudentTDistribution represents a continuous statistical distribution defined and supported over the set of real numbers and parametrized by a real number μ (called a "location parameter") and by positive real numbers σ and ν (called a "scale parameter" and the "degrees of freedom", respectively), which together determine the overall behavior of its probability density function (PDF).