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based on an earlier version of the Wolfram Language.
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# NoncentralStudentTDistribution

 NoncentralStudentTDistribution represents a noncentral Student distribution with degrees of freedom and noncentrality parameter .
Probability density function:
Cumulative distribution function:
Mean and variance:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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 Scope   (7)
Generate a set of pseudorandom numbers that have the noncentral 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:
Skewness is defined for :
Kurtosis is defined for :
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function has a strong mode for negative values of parameter :
Quantile function:
 Applications   (1)
The weight, in grams, of a particular boxed cereal product is known to follow a normal distribution with unknown mean . A test is performed with null hypothesis and alternative hypothesis . Fifteen boxes were chosen at random with sample mean weight of 363 and standard deviation of 32:
Critical value of -statistic at the 5% level:
Hence, the -test does not reject the null hypothesis:
Compute the power of the test, given and . In this case the test statistic follows a NoncentralStudentTDistribution:
The power of the -test to reject the null hypothesis is low:
Plot the power of the test as a function of sample size:
Find the sample size required for the power of the test to be at least 80%:
Parameter influence on the CDF for each :
Relationships to other distributions:
Noncentral distribution can be obtained from NormalDistribution and ChiSquareDistribution:
NoncentralStudentTDistribution is not defined when is not a positive real number:
NoncentralStudentTDistribution is not defined when is not a real number:
The characteristic function of a noncentral Student distribution has no closed-form representation:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: