Continuous Distributions¶

API: Continuous Distributions

API

Classes
class	cExponential
	Generates random numbers from the exponential distribution. More...
class	cNormal
	Generates random numbers from the normal distribution. More...
class	cTruncNormal
	Generates random numbers from the truncated normal distribution. More...
class	cGamma
	Generates random numbers from the gamma distribution. More...
class	cBeta
	Generates random numbers from the beta distribution. More...
class	cErlang
	Generates random numbers from the Erlang distribution. More...
class	cChiSquare
	Generates random numbers from the chi-square distribution. More...
class	cStudentT
	Generates random numbers from Student's T distribution. More...
class	cCauchy
	Generates random numbers from the Cauchy distribution. More...
class	cTriang
	Generates random numbers from the triangular distribution. More...
class	cWeibull
	Generates random numbers from the Weibull distribution. More...
class	cParetoShifted
	Generates random numbers from the shifted Pareto distribution. More...

Functions
SIM_API double	uniform (cRNG *rng, double a, double b)
	Returns a random variate with uniform distribution in the range [a,b).
SimTime	uniform (cRNG *rng, SimTime a, SimTime b)
	SimTime version of uniform(cRNG*,double,double), for convenience.
SIM_API double	exponential (cRNG *rng, double mean)
	Returns a random variate from the exponential distribution with the given mean (that is, with parameter lambda=1/mean).
SimTime	exponential (cRNG *rng, SimTime mean)
	SimTime version of exponential(cRNG*,double), for convenience.
SIM_API double	normal (cRNG *rng, double mean, double stddev)
	Returns a random variate from the normal distribution with the given mean and standard deviation.
SimTime	normal (cRNG *rng, SimTime mean, SimTime stddev)
	SimTime version of normal(cRNG*,double,double), for convenience.
SIM_API double	truncnormal (cRNG *rng, double mean, double stddev)
	Normal distribution truncated to nonnegative values.
SimTime	truncnormal (cRNG *rng, SimTime mean, SimTime stddev)
	SimTime version of truncnormal(cRNG*,double,double), for convenience.
SIM_API double	gamma_d (cRNG *rng, double alpha, double theta)
	Returns a random variate from the gamma distribution with parameters alpha>0, theta>0. Alpha is known as the "shape" parameter, and theta as the "scale" parameter.
SIM_API double	beta (cRNG *rng, double alpha1, double alpha2)
	Returns a random variate from the beta distribution with parameters alpha1, alpha2.
SIM_API double	erlang_k (cRNG *rng, unsigned int k, double mean)
	Returns a random variate from the Erlang distribution with k phases and mean mean.
SIM_API double	chi_square (cRNG *rng, unsigned int k)
	Returns a random variate from the chi-square distribution with k degrees of freedom.
SIM_API double	student_t (cRNG *rng, unsigned int i)
	Returns a random variate from the student-t distribution with i degrees of freedom. If Y1 has a normal distribution and Y2 has a chi-square distribution with k degrees of freedom then X = Y1 / sqrt(Y2/k) has a student-t distribution with k degrees of freedom.
SIM_API double	cauchy (cRNG *rng, double a, double b)
	Returns a random variate from the Cauchy distribution (also called Lorentzian distribution) with parameters a,b where b>0.
SIM_API double	triang (cRNG *rng, double a, double b, double c)
	Returns a random variate from the triangular distribution with parameters a <= b <= c.
double	lognormal (cRNG *rng, double m, double w)
	Returns a random variate from the lognormal distribution with "scale" parameter m and "shape" parameter w. m and w correspond to the parameters of the underlying normal distribution (m: mean, w: standard deviation.)
SIM_API double	weibull (cRNG *rng, double a, double b)
	Returns a random variate from the Weibull distribution with parameters a, b > 0, where a is the "scale" parameter and b is the "shape" parameter. Sometimes Weibull is given with alpha and beta parameters, then alpha=b and beta=a.
SIM_API double	pareto_shifted (cRNG *rng, double a, double b, double c)
	Returns a random variate from the shifted generalized Pareto distribution.

Description

Random variate generators for continuous distributions.

Function Documentation

◆ uniform() [1/2]

SIM_API double uniform	(	cRNG *	rng,
		double	a,
		double	b )

Returns a random variate with uniform distribution in the range [a,b).

Parameters

a,b	the interval, a<b
rng	the underlying random number generator

Referenced by cComponent::uniform(), and uniform().

◆ uniform() [2/2]

SimTime uniform	(	cRNG *	rng,
		SimTime	a,
		SimTime	b )

inline

SimTime version of uniform(cRNG*,double,double), for convenience.

References SimTime::dbl(), and uniform().

◆ exponential() [1/2]

SIM_API double exponential	(	cRNG *	rng,
		double	mean )

Returns a random variate from the exponential distribution with the given mean (that is, with parameter lambda=1/mean).

Parameters

mean	mean value
rng	the underlying random number generator

Referenced by cComponent::exponential(), and exponential().

◆ exponential() [2/2]

SimTime exponential	(	cRNG *	rng,
		SimTime	mean )

inline

SimTime version of exponential(cRNG*,double), for convenience.

References SimTime::dbl(), and exponential().

◆ normal() [1/2]

SIM_API double normal	(	cRNG *	rng,
		double	mean,
		double	stddev )

Returns a random variate from the normal distribution with the given mean and standard deviation.

Parameters

mean	mean of the normal distribution
stddev	standard deviation of the normal distribution
rng	the underlying random number generator

Referenced by lognormal(), cComponent::normal(), and normal().

◆ normal() [2/2]

SimTime normal	(	cRNG *	rng,
		SimTime	mean,
		SimTime	stddev )

inline

SimTime version of normal(cRNG*,double,double), for convenience.

References SimTime::dbl(), and normal().

◆ truncnormal() [1/2]

SIM_API double truncnormal	(	cRNG *	rng,
		double	mean,
		double	stddev )

Normal distribution truncated to nonnegative values.

It is implemented with a loop that discards negative values until a nonnegative one comes. This means that the execution time is not bounded: a large negative mean with much smaller stddev is likely to result in a large number of iterations.

The mean and stddev parameters serve as parameters to the normal distribution before truncation. The actual random variate returned will have a different mean and standard deviation.

Parameters

mean	mean of the normal distribution
stddev	standard deviation of the normal distribution
rng	the underlying random number generator

Referenced by cComponent::truncnormal(), and truncnormal().

◆ truncnormal() [2/2]

SimTime truncnormal	(	cRNG *	rng,
		SimTime	mean,
		SimTime	stddev )

inline

SimTime version of truncnormal(cRNG*,double,double), for convenience.

References SimTime::dbl(), and truncnormal().

◆ gamma_d()

SIM_API double gamma_d	(	cRNG *	rng,
		double	alpha,
		double	theta )

Returns a random variate from the gamma distribution with parameters alpha>0, theta>0. Alpha is known as the "shape" parameter, and theta as the "scale" parameter.

Some sources in the literature use the inverse scale parameter beta = 1 / theta, called the "rate" parameter. Various other notations can be found in the literature; our usage of (alpha,theta) is consistent with Wikipedia and Mathematica (Wolfram Research).

Gamma is the generalization of the Erlang distribution for non-integer k values, which becomes the alpha parameter. The chi-square distribution is a special case of the gamma distribution.

For alpha=1, Gamma becomes the exponential distribution with mean=theta.

The mean of this distribution is alpha*theta, and variance is alpha*theta².

Generation: if alpha=1, it is generated as exponential(theta).

For alpha>1, we make use of the acceptance-rejection method in "A Simple Method for Generating Gamma Variables", George Marsaglia and Wai Wan Tsang, ACM Transactions on Mathematical Software, Vol. 26, No. 3, September 2000.

The alpha<1 case makes use of the alpha>1 algorithm, as suggested by the above paper.

Remarks: the name gamma_d is chosen to avoid ambiguity with a function of the same name

Parameters

alpha	>0 the "shape" parameter
theta	>0 the "scale" parameter
rng	the underlying random number generator

Referenced by cComponent::gamma_d().

◆ beta()

SIM_API double beta	(	cRNG *	rng,
		double	alpha1,
		double	alpha2 )

Returns a random variate from the beta distribution with parameters alpha1, alpha2.

Generation is using relationship to Gamma distribution: if Y1 has gamma distribution with alpha=alpha1 and beta=1 and Y2 has gamma distribution with alpha=alpha2 and beta=2, then Y = Y1/(Y1+Y2) has beta distribution with parameters alpha1 and alpha2.

Parameters

alpha1,alpha2	>0
rng	the underlying random number generator

Referenced by cComponent::beta().

◆ erlang_k()

SIM_API double erlang_k	(	cRNG *	rng,
		unsigned int	k,
		double	mean )

Returns a random variate from the Erlang distribution with k phases and mean mean.

This is the sum of k mutually independent random variables, each with exponential distribution. Thus, the kth arrival time in the Poisson process follows the Erlang distribution.

Erlang with parameters m and k is gamma-distributed with alpha=k and beta=m/k.

Generation makes use of the fact that exponential distributions sum up to Erlang.

Parameters

k	number of phases, k>0
mean	>0
rng	the underlying random number generator

Referenced by cComponent::erlang_k().

◆ chi_square()

SIM_API double chi_square	(	cRNG *	rng,
		unsigned int	k )

Returns a random variate from the chi-square distribution with k degrees of freedom.

The chi-square distribution arises in statistics. If Yi are k independent random variates from the normal distribution with unit variance, then the sum-of-squares (sum(Yi^2)) has a chi-square distribution with k degrees of freedom.

The expected value of this distribution is k. Chi_square with parameter k is gamma-distributed with alpha=k/2, beta=2.

Generation is using relationship to gamma distribution.

Parameters

k	degrees of freedom, k>0
rng	the underlying random number generator

Referenced by cComponent::chi_square().

◆ student_t()

SIM_API double student_t	(	cRNG *	rng,
		unsigned int	i )

Returns a random variate from the student-t distribution with i degrees of freedom. If Y1 has a normal distribution and Y2 has a chi-square distribution with k degrees of freedom then X = Y1 / sqrt(Y2/k) has a student-t distribution with k degrees of freedom.

Generation is using relationship to gamma and chi-square.

Parameters

i	degrees of freedom, i>0
rng	the underlying random number generator

Referenced by cComponent::student_t().

◆ cauchy()

SIM_API double cauchy	(	cRNG *	rng,
		double	a,
		double	b )

Returns a random variate from the Cauchy distribution (also called Lorentzian distribution) with parameters a,b where b>0.

This is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.

Generation uses inverse transform.

Parameters

a
b	b>0
rng	the underlying random number generator

Referenced by cComponent::cauchy().

◆ triang()

SIM_API double triang	(	cRNG *	rng,
		double	a,
		double	b,
		double	c )

Returns a random variate from the triangular distribution with parameters a <= b <= c.

Generation uses inverse transform.

Parameters

a,b,c	a <= b <= c
rng	the underlying random number generator

Referenced by cComponent::triang().

◆ lognormal()

double lognormal	(	cRNG *	rng,
		double	m,
		double	w )

inline

Returns a random variate from the lognormal distribution with "scale" parameter m and "shape" parameter w. m and w correspond to the parameters of the underlying normal distribution (m: mean, w: standard deviation.)

Generation is using relationship to normal distribution.

Parameters

m	"scale" parameter, m>0
w	"shape" parameter, w>0
rng	the underlying random number generator

References normal().

Referenced by cComponent::lognormal().

◆ weibull()

SIM_API double weibull	(	cRNG *	rng,
		double	a,
		double	b )

Returns a random variate from the Weibull distribution with parameters a, b > 0, where a is the "scale" parameter and b is the "shape" parameter. Sometimes Weibull is given with alpha and beta parameters, then alpha=b and beta=a.

The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed to quantify fatigue data, but it is also used in reliability analysis of systems involving a "weakest link," e.g. in calculating a device's mean time to failure.

When b=1, Weibull(a,b) is exponential with mean a.

Generation uses inverse transform.

Parameters

a	the "scale" parameter, a>0
b	the "shape" parameter, b>0
rng	the underlying random number generator

Referenced by cComponent::weibull().

◆ pareto_shifted()

SIM_API double pareto_shifted	(	cRNG *	rng,
		double	a,
		double	b,
		double	c )

Returns a random variate from the shifted generalized Pareto distribution.

Generation uses inverse transform.

Parameters

a,b	the usual parameters for generalized Pareto
c	shift parameter for left-shift
rng	the underlying random number generator

Referenced by cComponent::pareto_shifted().

Continuous Distributions¶

Classes

Functions

Description

Function Documentation

◆ uniform() [1/2]

◆ uniform() [2/2]

◆ exponential() [1/2]

◆ exponential() [2/2]

◆ normal() [1/2]

◆ normal() [2/2]

◆ truncnormal() [1/2]

◆ truncnormal() [2/2]

◆ gamma_d()

◆ beta()

◆ erlang_k()

◆ chi_square()

◆ student_t()

◆ cauchy()

◆ triang()

◆ lognormal()

◆ weibull()

◆ pareto_shifted()