# SuanShu, a Java numerical and statistical library

com.numericalmethod.suanshu.stats.distribution.univariate

## Class TDistribution

• java.lang.Object
• com.numericalmethod.suanshu.stats.distribution.univariate.TDistribution
• All Implemented Interfaces:
ProbabilityDistribution

public class TDistribution
extends Object
implements ProbabilityDistribution
The Student t distribution is the probability distribution of t, where $t = \frac{\bar{x} - \mu}{s / \sqrt N}$
• $$\bar{x}$$ is the sample mean;
• μ is the population mean;
• s is the square root of the sample variance;
• N is the sample size;
The importance of the Student's distribution is when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data. This is especially true when the sample size is small. When the sample size is large, the Student's distribution converges to the Normal distribution.

The R equivalent functions are dt, pt, qt, rt.

Wikipedia: Student's t-distribution
• ### Constructor Summary

Constructors
Constructor and Description
TDistribution(double v)
Construct a Student's t distribution.
• ### Method Summary

All Methods
Modifier and Type Method and Description
double cdf(double x)
Gets the cumulative probability F(x) = Pr(X ≤ x).
double density(double x)
The density function, which, if exists, is the derivative of F.
double entropy()
Gets the entropy of this distribution.
double kurtosis()
Gets the excess kurtosis of this distribution.
double mean()
Gets the mean of this distribution.
double median()
Gets the median of this distribution.
double moment(double x)
The moment generating function is the expected value of etX.
double quantile(double u)
Gets the quantile, the inverse of the cumulative distribution function.
double skew()
Gets the skewness of this distribution.
double variance()
Gets the variance of this distribution.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### TDistribution

public TDistribution(double v)
Construct a Student's t distribution.
Parameters:
v - the degree of freedom
• ### Method Detail

• #### mean

public double mean()
Gets the mean of this distribution.
Specified by:
mean in interface ProbabilityDistribution
Returns:
the mean
Throws:
UnsupportedOperationException - when v <= 1
Wikipedia: Expected value
• #### median

public double median()
Description copied from interface: ProbabilityDistribution
Gets the median of this distribution.
Specified by:
median in interface ProbabilityDistribution
Returns:
the median
Wikipedia: Median
• #### variance

public double variance()
Gets the variance of this distribution.
Specified by:
variance in interface ProbabilityDistribution
Returns:
the variance
Throws:
UnsupportedOperationException - when v < 2
Wikipedia: Variance
• #### skew

public double skew()
Gets the skewness of this distribution.
Specified by:
skew in interface ProbabilityDistribution
Returns:
the skewness
Throws:
UnsupportedOperationException - when v <= 3
Wikipedia: Skewness
• #### kurtosis

public double kurtosis()
Gets the excess kurtosis of this distribution.
Specified by:
kurtosis in interface ProbabilityDistribution
Returns:
the excess kurtosis
Throws:
UnsupportedOperationException - when v <= 4
Wikipedia: Kurtosis
• #### entropy

public double entropy()
Description copied from interface: ProbabilityDistribution
Gets the entropy of this distribution.
Specified by:
entropy in interface ProbabilityDistribution
Returns:
the entropy
Wikipedia: Entropy (information theory)
• #### cdf

public double cdf(double x)
Description copied from interface: ProbabilityDistribution
Gets the cumulative probability F(x) = Pr(X ≤ x).
Specified by:
cdf in interface ProbabilityDistribution
Parameters:
x - x
Returns:
F(x) = Pr(X ≤ x)
Wikipedia: Cumulative distribution function
• #### density

public double density(double x)
Description copied from interface: ProbabilityDistribution
The density function, which, if exists, is the derivative of F. It describes the density of probability at each point in the sample space.
f(x) = dF(X) / dx
This may not always exist.

For the discrete cases, this is the probability mass function. It gives the probability that a discrete random variable is exactly equal to some value.

Specified by:
density in interface ProbabilityDistribution
Parameters:
x - x
Returns:
f(x)
• #### quantile

public double quantile(double u)
Description copied from interface: ProbabilityDistribution
Gets the quantile, the inverse of the cumulative distribution function. It is the value below which random draws from the distribution would fall u×100 percent of the time.

F-1(u) = x, such that
Pr(X ≤ x) = u

This may not always exist.
Specified by:
quantile in interface ProbabilityDistribution
Parameters:
u - u, a quantile
Returns:
F-1(u)
Wikipedia: Quantile function
• #### moment

public double moment(double x)
Description copied from interface: ProbabilityDistribution
The moment generating function is the expected value of etX. That is,
E(etX)
This may not always exist.
Specified by:
moment in interface ProbabilityDistribution
Parameters:
x - t
Returns:
E(exp(tX))