com.numericalmethod.suanshu.stats.distribution.univariate

Class NormalDistribution

java.lang.Object

com.numericalmethod.suanshu.stats.distribution.univariate.NormalDistribution

All Implemented Interfaces:

ProbabilityDistribution, java.io.Serializable

public class NormalDistribution
extends java.lang.Object
implements ProbabilityDistribution

The Normal distribution has its density a Gaussian function. The Normal distribution is probably the most important single distribution. By the central limit theorem, under certain conditions, the sum of a number of random variables with finite means and variances approaches a Normal distribution as the number of variables increases. Laplace proved that the Normal distribution occurs as a limiting distribution of arithmetic means of independent, identically distributed random variables with finite second moment.

The R equivalent functions are dnorm, pnorm, qnorm, rnorm.

See Also:

• Wikipedia: NormalDistribution distribution
• Gaussian

, Serialized Form

Constructor Summary

Constructors

Constructor and Description
NormalDistribution() Construct an instance of the standard Normal distribution with mean 0 and standard deviation 1.
NormalDistribution(double mu, double sigma) Construct a Normal distribution with mean mu and standard deviation sigma.

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 t) 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

NormalDistribution

public NormalDistribution()

Construct an instance of the standard Normal distribution with mean 0 and standard deviation 1.

NormalDistribution

public NormalDistribution(double mu,
                          double sigma)

Construct a Normal distribution with mean mu and standard deviation sigma.

Parameters:

mu - the mean

sigma - the standard deviation

Method Detail

mean

public double mean()

Description copied from interface: ProbabilityDistribution

Get the mean of this distribution.

Specified by:

mean in interface ProbabilityDistribution

Returns:

the mean

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()

Description copied from interface: ProbabilityDistribution

Get the variance of this distribution.

Specified by:

variance in interface ProbabilityDistribution

Returns:

the variance

See Also:

Wikipedia: Variance

skew

public double skew()

Description copied from interface: ProbabilityDistribution

Get the skewness of this distribution.

Specified by:

skew in interface ProbabilityDistribution

Returns:

the skewness

See Also:

Wikipedia: Skewness

kurtosis

public double kurtosis()

Description copied from interface: ProbabilityDistribution

Get the excess kurtosis of this distribution.

Specified by:

kurtosis in interface ProbabilityDistribution

Returns:

the excess kurtosis

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

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

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

moment

public double moment(double t)

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:

t - x

Returns:

E(exp(tX))

See Also:

Wikipedia: Moment-generating function