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# StudentTDistribution

 StudentTDistribution[] represents a Student distribution with degrees of freedom. StudentTDistributionrepresents a Student distribution with location parameter , scale parameter , and degrees of freedom.
• The probability density for value in a Student distribution with degrees of freedom is proportional to . »
• With location parameter and scale , follows a standard Student distribution with degrees of freedom.
• For integer , the Student distribution gives the distribution of the deviation from the true mean of the observed mean for a sample of values from a normal distribution, normalized by standard deviation of the sample.
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function of a generalized Student distribution:
Cumulative distribution function of a generalized Student distribution:
Mean and variance of a generalized Student distribution:
Median of a generalized Student distribution:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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Probability density function of a generalized Student distribution:
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Cumulative distribution function of a generalized Student distribution:
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Mean and variance of a generalized Student distribution:
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Median of a generalized Student distribution:
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 Scope   (7)
Generate a set of random numbers that have the Student distribution:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
A Student distribution is symmetric and hence skewness is 0 if defined:
Kurtosis:
Adding scale and location parameters does not change the kurtosis:
In the limit, kurtosis is the same as for NormalDistribution:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Moment for generalized Student distribution:
Closed form for symbolic order:
CentralMoment for generalized Student distribution:
Closed form for symbolic order:
FactorialMoment for generalized Student distribution:
Cumulant for generalized Student distribution:
Hazard function:
For generalized Student distribution:
Quantile function:
For generalized Student distribution:
 Applications   (2)
Compute -values for a -test with degrees of freedom and alternative hypothesis :
Alternative hypothesis :
Alternative hypothesis :
StudentTDistribution is used in exact (small) sampling theory. Define -statistics:
If data comes from a normal distribution, then the -statistics follow a StudentTDistribution, even for data being a sample of small size (less than 30):
Parameter influence on the CDF for each :
Student distribution is closed under translation and scaling by a positive factor:
StudentTDistribution[] converges to a normal distribution as :
Relationships to other distributions:
StudentTDistribution[] has location and scale :
The two forms are related by a change of variable:
The Student distribution converges to the standard NormalDistribution as tends to infinity:
A square of a Student distributed variable has FRatioDistribution:
An inverse square of Student distributed variable has FRatioDistribution:
Student distribution is a special case of type 4 and type 7 PearsonDistribution:
Generalized Student distribution is a special case of type 4 and type 7 PearsonDistribution:
Student distribution can be obtained from ChiSquareDistribution:
Student distribution can be obtained from NormalDistribution and ChiSquareDistribution:
Marginals of MultivariateTDistribution with identity scale matrix are Student distributions:
Central moments of two Student distributions are proportional when defined:
StudentTDistribution is not defined when is a not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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