#include <boost/math/distributions/students_t.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class students_t_distribution; typedef students_t_distribution<> students_t; template <class RealType, class Policy> class students_t_distribution { typedef RealType value_type; typedef Policy policy_type; // Construct: students_t_distribution(const RealType& v); // Accessor: RealType degrees_of_freedom()const; // degrees of freedom estimation: static RealType find_degrees_of_freedom( RealType difference_from_mean, RealType alpha, RealType beta, RealType sd, RealType hint = 100); }; }} // namespaces A statistical distribution published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose "Student". Given N independent measurements, let
where M is the population mean,μ is the sample mean, and s is the sample variance. Student's t-distribution is defined as the distribution of the random variable t which is - very loosely - the "best" that we can do not knowing the true standard deviation of the sample. It has the PDF:
The Student's t-distribution takes a single parameter: the number of degrees of freedom of the sample. When the degrees of freedom is one then this distribution is the same as the Cauchy-distribution. As the number of degrees of freedom tends towards infinity, then this distribution approaches the normal-distribution. The following graph illustrates how the PDF varies with the degrees of freedom ν:
Member Functionsstudents_t_distribution(const RealType& v); Constructs a Student's t-distribution with v degrees of freedom. Requires v > 0, otherwise calls domain_error. Note that non-integral degrees of freedom are supported, and meaningful under certain circumstances. RealType degrees_of_freedom()const; Returns the number of degrees of freedom of this distribution. static RealType find_degrees_of_freedom( RealType difference_from_mean, RealType alpha, RealType beta, RealType sd, RealType hint = 100); Returns the number of degrees of freedom required to observe a significant result in the Student's t test when the mean differs from the "true" mean by difference_from_mean.
For more information on this function see the NIST Engineering Statistics Handbook. Non-member AccessorsAll the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support. The domain of the random variable is [-∞, +∞]. ExamplesVarious worked examples are available illustrating the use of the Student's t distribution. AccuracyThe normal distribution is implemented in terms of the incomplete beta function and it's inverses, refer to accuracy data on those functions for more information. ImplementationIn the following table v is the degrees of freedom of the distribution, t is the random variate, p is the probability and q = 1-p.
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