Extends the class ContinuousDistribution for the normal distribution (e.g., [99] (page 80)). More...

Inheritance diagram for NormalDist:

Collaboration diagram for NormalDist:

Public Member Functions
	NormalDist ()
	Constructs a `NormalDist` object with default parameters \(\mu= 0\) and \(\sigma= 1\).

	NormalDist (double mu, double sigma)
	Constructs a `NormalDist` object with mean \(\mu\) = `mu` and standard deviation \(\sigma\) = `sigma`.

double	density (double x)

double	cdf (double x)
	Returns the distribution function \(F(x)\). More...

double	barF (double x)
	Returns \(\bar{F}(x) = 1 - F(x)\). More...

double	inverseF (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More...

double	getMean ()
	Returns the mean of the distribution function.

double	getVariance ()
	Returns the variance of the distribution function.

double	getStandardDeviation ()
	Returns the standard deviation of the distribution function.

double	getMu ()
	Returns the parameter \(\mu\).

double	getSigma ()
	Returns the parameter \(\sigma\).

void	setParams (double mu, double sigma)
	Sets the parameters \(\mu\) and \(\sigma\) of this object.

double []	getParams ()
	Return a table containing the parameters of the current distribution. More...

String	toString ()
	Returns a `String` containing information about the current distribution.

Public Member Functions inherited from ContinuousDistribution
abstract double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\). More...

double	barF (double x)
	Returns the complementary distribution function. More...

double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...

double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...

double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\). More...

double	getMean ()
	Returns the mean. More...

double	getVariance ()
	Returns the variance. More...

double	getStandardDeviation ()
	Returns the standard deviation. More...

double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXinf (double xa)
	Sets the value \(x_a=\) `xa`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXsup (double xb)
	Sets the value \(x_b=\) `xb`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

Static Public Member Functions
static double	density01 (double x)
	Same as density(0, 1, x).

static double	density (double mu, double sigma, double x)
	Computes the normal density function ( fnormal ).

static double	cdf01 (double x)
	Same as cdf(0, 1, x).

static double	cdf (double mu, double sigma, double x)
	Computes the normal distribution function with mean \(\mu\) and variance \(\sigma^2\). More...

static double	barF01 (double x)
	Same as barF(0, 1, x).

static double	barF (double mu, double sigma, double x)
	Computes the complementary normal distribution function \(\bar{F}(x) = 1 - \Phi((x-\mu)/\sigma)\), with mean \(\mu\) and variance \(\sigma^2\). More...

static double	inverseF01 (double u)
	Same as inverseF(0, 1, u).

static double	inverseF (double mu, double sigma, double u)
	Computes the inverse normal distribution function with mean \(\mu\) and variance \(\sigma^2\). More...

static double []	getMLE (double[] x, int n)
	Estimates the parameters \((\mu, \sigma)\) of the normal distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...

static NormalDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a normal distribution with parameters \(\mu\) and \(\sigma\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static double	getMean (double mu, double sigma)
	Computes and returns the mean \(E[X] = \mu\) of the normal distribution with parameters \(\mu\) and \(\sigma\). More...

static double	getVariance (double mu, double sigma)
	Computes and returns the variance \(\mbox{Var}[X] = \sigma^2\) of the normal distribution with parameters \(\mu\) and \(\sigma\). More...

static double	getStandardDeviation (double mu, double sigma)
	Computes and returns the standard deviation \(\sigma\) of the normal distribution with parameters \(\mu\) and \(\sigma\). More...

Protected Attributes
double	mu

double	sigma

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

Static Protected Attributes
static final double	RAC2PI = 2.50662827463100050

Static Protected Attributes inherited from ContinuousDistribution
static final double	XBIG = 100.0

static final double	XBIGM = 1000.0

static final double []	EPSARRAY

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

Detailed Description

Extends the class ContinuousDistribution for the normal distribution (e.g., [99] (page 80)).

It has mean \(\mu\) and variance \(\sigma^2\). Its density function is

\[ f (x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)} \qquad\mbox{for } -\infty< x < \infty, \tag{fnormal} \]

where \(\sigma> 0\). When \(\mu=0\) and \(\sigma=1\), we have the standard normal distribution, with corresponding distribution function

\[ F(x) = \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} dt \qquad\mbox{for } -\infty< x < \infty. \tag{cdfnormal} \]

The non-static methods cdf, barF, and inverseF are implemented via cdf01, barF01, and inverseF01, respectively.

Member Function Documentation

◆ barF() [1/2]

double barF ( double x )

Returns \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Implements Distribution.

◆ barF() [2/2]

static double barF	(	double	mu,
		double	sigma,
		double	x
	)

static

Computes the complementary normal distribution function \(\bar{F}(x) = 1 - \Phi((x-\mu)/\sigma)\), with mean \(\mu\) and variance \(\sigma^2\).

Uses a Chebyshev series giving 16 decimal digits of precision [212] .

◆ cdf() [1/2]

double cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements Distribution.

◆ cdf() [2/2]

static double cdf	(	double	mu,
		double	sigma,
		double	x
	)

static

Computes the normal distribution function with mean \(\mu\) and variance \(\sigma^2\).

Uses the Chebyshev approximation proposed in [212] , which gives 16 decimals of precision.

◆ getInstanceFromMLE()

static NormalDist getInstanceFromMLE	(	double []	x,
		int	n
	)

static

Creates a new instance of a normal distribution with parameters \(\mu\) and \(\sigma\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).

Parameters

x	the list of observations to use to evaluate parameters
n	the number of observations to use to evaluate parameters

◆ getMean()

static double getMean	(	double	mu,
		double	sigma
	)

static

Computes and returns the mean \(E[X] = \mu\) of the normal distribution with parameters \(\mu\) and \(\sigma\).

Returns: the mean of the normal distribution \(E[X] = \mu\)

◆ getMLE()

static double [] getMLE	(	double []	x,
		int	n
	)

static

Estimates the parameters \((\mu, \sigma)\) of the normal distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).

The estimates are returned in a two-element array, in regular order: [ \(\hat{\mu}\), \(\hat{\sigma}\)]. The maximum likelihood estimators are the values \((\hat{\mu}, \hat{\sigma})\) that satisfy the equations:

\begin{align*} \hat{\mu} & = \bar{x}_n = \frac{1}{n} \sum_{i=1}^n x_i \\ \hat{\sigma} & = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x}_n)^2}. \end{align*}

See [99] (page 123).

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters

Returns: returns the parameters [ \(\hat{\mu}\), \(\hat{\sigma}\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\mu\), \(\sigma\)].

Implements Distribution.

◆ getStandardDeviation()

static double getStandardDeviation	(	double	mu,
		double	sigma
	)

static

Computes and returns the standard deviation \(\sigma\) of the normal distribution with parameters \(\mu\) and \(\sigma\).

Returns: the standard deviation of the normal distribution

◆ getVariance()

static double getVariance	(	double	mu,
		double	sigma
	)

static

Computes and returns the variance \(\mbox{Var}[X] = \sigma^2\) of the normal distribution with parameters \(\mu\) and \(\sigma\).

Returns: the variance of the normal distribution \(\mbox{Var}[X] = \sigma^2\)

◆ inverseF() [1/2]

double inverseF ( double u )

Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).

Parameters

u	value in the interval \((0,1)\) for which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Implements Distribution.

◆ inverseF() [2/2]

static double inverseF	(	double	mu,
		double	sigma,
		double	u
	)

static

Computes the inverse normal distribution function with mean \(\mu\) and variance \(\sigma^2\).

Uses different rational Chebyshev approximations [21] . Returns 16 decimal digits of precision for \(2.2\times10^{-308} < u < 1\).

The documentation for this class was generated from the following file:

NormalDist.java

Public Member Functions

Static Public Member Functions

Protected Attributes

Static Protected Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getParams()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF() [1/2]

◆ inverseF() [2/2]