NoncentralStudentTDistribution

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:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function has a strong mode for negative values of parameter :

Quantile function:

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:

NoncentralStudentTDistribution is equivalent to StudentTDistribution[]:

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: