com.numericalmethod.suanshu.stats.distribution.univariate

Class TDistribution

java.lang.Object

com.numericalmethod.suanshu.stats.distribution.univariate.TDistribution

All Implemented Interfaces:

ProbabilityDistribution, java.io.Serializable

public class TDistribution
extends java.lang.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.

See Also:

Wikipedia: Student's t-distribution, Serialized Form

Constructor Summary

Constructors

Constructor and Description
TDistribution(double v) Construct a Student's t distribution.

Method Summary

Modifier and Type	Method and Description
double	cdf(double x) Get 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() Get the entropy of this distribution.
double	kurtosis() Get the excess kurtosis of this distribution.
double	mean() Get the mean of this distribution.
double	median() Get the median of this distribution.
double	moment(double x) The moment generating function is the expected value of etX.
double	quantile(double u) Get the quantile, the inverse of the cumulative distribution function.
double	skew() Get the skewness of this distribution.
double	variance() Get 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()

Get the mean of this distribution.

Specified by:

mean in interface ProbabilityDistribution

Returns:

the mean

Throws:

java.lang.UnsupportedOperationException - when v <= 1

See Also:

Wikipedia: Expected value

median

public double median()

Description copied from interface: ProbabilityDistribution

Get the median of this distribution.

Specified by:

median in interface ProbabilityDistribution

Returns:

the median

See Also:

Wikipedia: Median

variance

public double variance()

Get the variance of this distribution.

Specified by:

variance in interface ProbabilityDistribution

Returns:

the variance

Throws:

java.lang.UnsupportedOperationException - when v < 2

See Also:

Wikipedia: Variance

skew

public double skew()

Get the skewness of this distribution.

Specified by:

skew in interface ProbabilityDistribution

Returns:

the skewness

Throws:

java.lang.UnsupportedOperationException - when v <= 3

See Also:

Wikipedia: Skewness

kurtosis

public double kurtosis()

Get the excess kurtosis of this distribution.

Specified by:

kurtosis in interface ProbabilityDistribution

Returns:

the excess kurtosis

Throws:

java.lang.UnsupportedOperationException - when v <= 4

See Also:

Wikipedia: Kurtosis

entropy

public double entropy()

Description copied from interface: ProbabilityDistribution

Get the entropy of this distribution.

Specified by:

entropy in interface ProbabilityDistribution

Returns:

the entropy

See Also:

Wikipedia: Entropy (information theory)

cdf

public double cdf(double x)

Description copied from interface: ProbabilityDistribution

Get the cumulative probability F(x) = Pr(X ≤ x).

Specified by:

cdf in interface ProbabilityDistribution

Parameters:

x - x

Returns:

F(x) = Pr(X ≤ x)

See Also:

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)

See Also:

• Wikipedia: Probability density function
• Wikipedia: Probability mass function

quantile

public double quantile(double u)

Description copied from interface: ProbabilityDistribution

Get 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)

See Also:

Wikipedia: Quantile function

moment

public double moment(double x)

Description copied from interface: ProbabilityDistribution

The moment generating function is the expected value of e^tX. That is,

E(e^tX)

This may not always exist.

Specified by:

moment in interface ProbabilityDistribution

Parameters:

x - x

Returns:

E(exp(tX))

See Also:

Wikipedia: Moment-generating function