# SuanShu, a Java numerical and statistical library

com.numericalmethod.suanshu.stats.distribution.univariate

## Class NormalDistribution

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

public class NormalDistribution
extends 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.

• ### 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

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 t)
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

• #### 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
Gets the mean of this distribution.
Specified by:
mean in interface ProbabilityDistribution
Returns:
the mean
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()
Description copied from interface: ProbabilityDistribution
Gets the variance of this distribution.
Specified by:
variance in interface ProbabilityDistribution
Returns:
the variance
Wikipedia: Variance
• #### skew

public double skew()
Description copied from interface: ProbabilityDistribution
Gets the skewness of this distribution.
Specified by:
skew in interface ProbabilityDistribution
Returns:
the skewness
Wikipedia: Skewness
• #### kurtosis

public double kurtosis()
Description copied from interface: ProbabilityDistribution
Gets the excess kurtosis of this distribution.
Specified by:
kurtosis in interface ProbabilityDistribution
Returns:
the excess kurtosis
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
• #### 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
• #### 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)
• #### moment

public double moment(double t)
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:
t - t
Returns:
E(exp(tX))