Package com.imsl.stat

Class InvCdf

java.lang.Object

com.imsl.stat.InvCdf

public final class InvCdf
extends Object

Inverse cumulative probability distribution functions.

See Also:

• Example
• Cumulative distribution function
• Probability density function

Method Summary

Modifier and Type	Method	Description
static double	beta(double p, double pin, double qin)	Returns the inverse of the beta cumulative probability distribution function.
static double	chi(double p, double df)	Returns the inverse of the chi-squared cumulative probability distribution function.
static double	continuousUniform(double p, double a, double b)	Returns the inverse of the continuous uniform cumulative distribution function.
static int	discreteUniform(double p, int n)	Returns the inverse of the discrete uniform cumulative probability distribution function.
static double	exponential(double p, double scale)	Returns the inverse of the exponential cumulative probability distribution function.
static double	extremeValue(double p, double mu, double beta)	Returns the inverse of the extreme value cumulative probability distribution function.
static double	F(double p, double dfn, double dfd)	Returns the inverse of the F cumulative probability distribution function.
static double	gamma(double p, double a)	Returns the inverse of the gamma cumulative probability distribution function.
static double	generalizedExtremeValue(double p, double mu, double sigma, double xi)	Returns the inverse of the generalized extreme value cumulative distribution function.
static double	generalizedGaussian(double p, double mu, double alpha, double beta)	Returns the inverse of the generalized Gaussian (normal) cumulative distribution function.
static double	generalizedPareto(double p, double mu, double sigma, double alpha)	Returns the inverse of the generalized Pareto cumulative distribution function.
static double	geometric(double r, double pin)	Returns the inverse of the discrete geometric cumulative probability distribution function.
static double	logistic(double p, double mu, double sigma)	Returns the inverse of the logistic cumulative probability distribution function.
static double	logNormal(double p, double mu, double sigma)	Returns the inverse of the standard lognormal cumulative probability distribution function.
static double	noncentralBeta(double p, double shape1, double shape2, double lambda)	Returns the inverse of the noncentral beta cumulative distribution function (CDF).
static double	noncentralchi(double p, double df, double alam)	Returns the inverse of the noncentral chi-squared cumulative probability distribution function.
static double	noncentralF(double p, double dfn, double dfd, double lambda)	Returns the inverse of the noncentral F cumulative distribution function (CDF).
static double	noncentralstudentsT(double p, int idf, double delta)	Returns the inverse of the noncentral Student's t cumulative probability distribution function.
static double	normal(double p)	Returns the inverse of the normal (Gaussian) cumulative probability distribution function.
static double	Pareto(double p, double xm, double k)	Returns the inverse of the Pareto cumulative probability density function.
static double	Rayleigh(double p, double alpha)	Returns the inverse of the Rayleigh cumulative probability distribution function.
static double	studentsT(double p, double df)	Returns the inverse of the Student's t cumulative probability distribution function.
static double	uniform(double p, double aa, double bb)	Deprecated. use continuousUniform(double, double, double) instead
static double	Weibull(double p, double gamma, double alpha)	Returns the inverse of the Weibull cumulative probability distribution function.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

beta

public static double beta(double p,
 double pin,
 double qin)

Returns the inverse of the beta cumulative probability distribution function.

Method beta evaluates the inverse distribution function of a beta random variable with parameters pin and qin, that is, with P = p, p = pin, and q = qin, it determines x (equal to beta (p, pin, qin)), such that

$$P = \frac{{\Gamma \left( p \right)\Gamma \left( q \right)}}{{\Gamma \left( {p + q} \right)}}\int_0^x {t^{p - 1} } \left( {1 - t} \right)^{q - 1} dt$$

where \(\Gamma(\cdot)\) is the gamma function. The probability that the random variable takes a value less than or equal to x is P.

Parameters:

p - a double, the probability for which the inverse of the beta CDF is to be evaluated.

pin - a double, the first beta distribution parameter.

qin - a double, the second beta distribution parameter.

Returns:

a double, the pth quantile x given the parameter values.

See Also:

• Example

F

public static double F(double p,
 double dfn,
 double dfd)

Returns the inverse of the F cumulative probability distribution function.

Method F evaluates the inverse distribution function of a Snedecor's F random variable with dfn numerator degrees of freedom and dfd denominator degrees of freedom. The function is evaluated by making a transformation to a beta random variable and then using beta. If X is an F variate with \(v_1\) and \(v_2\) degrees of freedom and \(Y = v_1X/(v_2 + v_1X)\), then Y is a beta variate with parameters \(p = v_1/2\) and \(q = v_2/2\). If \(P \le 0.5\), F uses this relationship directly, otherwise, it also uses a relationship between X random variables that can be expressed as follows, using f, which is the F cumulative distribution function:

$${\rm F}(X, {\it dfn}, {\it dfd})=1 - {\rm F}(1/X, {\it dfd}, {\it dfn})$$

Parameters:

p - a double, the probability for which the inverse of the F distribution function is to be evaluated. Argument p must be in the open interval (0.0, 1.0).

dfn - a double, the numerator degrees of freedom. It must be positive.

dfd - a double, the denominator degrees of freedom. It must be positive.

Returns:

a double, the pth quantile x given the parameter values.

See Also:

• Example

gamma

public static double gamma(double p,
 double a)

Returns the inverse of the gamma cumulative probability distribution function.

Method gamma evaluates the inverse distribution function of a gamma random variable with shape parameter a. That is, it determines \(x \) such that

$$p = \frac{1}{{\Gamma \left( a \right)}}\int_o^x {e^{ - t} } t^{a - 1} dt$$

where \(\Gamma(\cdot)\) is the gamma function. The probability that the random variable takes a value less than or equal to x is p. See the documentation for routine gamma for further discussion of the gamma distribution.

gamma uses bisection and modified regula falsi to invert the distribution function, which is evaluated using method gamma.

Parameters:

p - a double, the probability at which the function is to be evaluated

a - a double, the value of the shape parameter. This must be positive.

Returns:

a double, the pth quantile x given the parameter values

See Also:

• Example

generalizedExtremeValue

public static double generalizedExtremeValue(double p,
 double mu,
 double sigma,
 double xi)

Returns the inverse of the generalized extreme value cumulative distribution function.

Returns the quantile \(x\) such that \(p = F(x|\mu,\sigma,\xi) \) where \(p\) is a probability and $$F(x|\mu,\sigma,\xi)=e^{-t(x)}$$ where $$t(x)=\left\{\begin{array}{lll} (1+\xi(\frac{x-\mu}{\sigma}))^{-\frac{1}{\xi}}& \mbox{for} & \xi \ne 0 \\ e^{-\frac{x-\mu}{\sigma}} & \mbox{for} & \xi = 0 \end{array}\right. $$ \( \mu\in \mathbb{R} \) is the location parameter, \(\sigma > 0 \) is the scale parameter, and \(\xi \in \mathbb{R} \) is the shape parameter. Furthermore, the support for the distribution is $$\left\{\begin{array}{lll} x \in \mathbb{R} & \mbox{for} & \xi = 0 \\ x \ge \mu - \frac{\sigma}{\xi} & \mbox{for} & \xi > 0 \\ x \le \mu - \frac{\sigma}{\xi} & \mbox{for} & \xi \lt 0 \end{array}\right. $$

Parameters:

p - a double, the probability at which to evaluate the inverse cdf

mu - a double, the location parameter

sigma - a double, the scale parameter

xi - a double, the shape parameter

Returns:

a double, the pth quantile x given the parameter values

generalizedGaussian

public static double generalizedGaussian(double p,
 double mu,
 double alpha,
 double beta)

Returns the inverse of the generalized Gaussian (normal) cumulative distribution function.

Returns the quantile \(x\) satisfying \(p=F(x)\), where \(p\) is a probability. That is, the function solves for x in $$ p = F(x;\mu,\alpha,\beta) $$ where $$F(x;\mu,\alpha,\beta)=\frac{1}{2} + \frac{sign(x-\mu)}{2\Gamma(\frac{1}{\beta})}\gamma(\frac{1}{\beta},(\frac{|x-\mu|}{\alpha})^{\beta})$$ where \(\mu\) is the location parameter, \(\alpha\) is the scale parameter, \(\beta\) is the shape parameter, and \(\gamma(a,t)\) is the incomplete gamma function with parameter \(a\) evaluated at \(t\). The solution can be shown to be $$ x= sign(p-0.5)\alpha g^{-1}(|2p-1|,\frac{1}{\beta})^{\frac{1}{\beta}} + \mu$$ where \(g^{-1}(u,a)\) is the inverse gamma cumulative distribution function with shape parameter \(a\) evaluated at \(u\).

Parameters:

p - a double, the probability at which the function is to be evaluated

mu - a double, the location parameter

alpha - a double, the scale parameter. It must be positive.

beta - a double, the shape parameter. It must be positive.

Returns:

a double, the pth quantile x given the parameter values

generalizedPareto

public static double generalizedPareto(double p,
 double mu,
 double sigma,
 double alpha)

Returns the inverse of the generalized Pareto cumulative distribution function.

Returns the quantile \(x\) satisfying \(p=F(x)\), where \(p\) is a probability. That is, the function solves for x in $$ p = F(x; \mu,\sigma, \alpha)$$ where $$ F(x; \mu,\sigma, \alpha) =\left\{\begin{array}{cl} 1 - \left( 1 + \frac{\alpha (x-\mu)}{\sigma}\right)^{(-\frac{1}{\alpha})} & \mbox{for}~ \alpha \ne 0 \\ 1 - \exp(-\frac{x-\mu}{\sigma}) & \mbox{for}~ \alpha = 0 \end{array}\right.$$ where \(\mu \in \mathbb{R} \) is the location parameter, \(\sigma \gt 0 \) is the scale parameter, and \(\alpha \in \mathbb{R} \) is the shape parameter. The support for \(\alpha \ge 0\) is \( x \ge \mu \), while for \( \alpha \lt 0 \), \(\mu \le x \le \mu - \frac{\sigma}{\alpha}\). The solution is easily shown to be $$ x =\left\{\begin{array}{cl} \left(\frac{(1 - p)^{-\alpha}-1}{\alpha}\right)\sigma +\mu & \mbox{for}~ \alpha \ne 0 \\ \mu - \sigma\ln(1-p) & \mbox{for}~ \alpha = 0 \end{array}\right.$$ Note that this follows the parameterization given in Wikipedia.

References
1. Wikipedia contributors. "Generalized Pareto distribution." Wikipedia, The Free Encyclopedia.

Parameters:

p - a double, the probability at which the function is to be evaluated

mu - a double, the location parameter

sigma - a double, the scale parameter. It must be positive.

alpha - a double, the shape parameter

Returns:

a double, the probability density at x given the parameter values

exponential

public static double exponential(double p,
 double scale)

Returns the inverse of the exponential cumulative probability distribution function.

Method exponential evaluates the inverse distribution function of a gamma random variable with scale parameter b and shape parameter a=1.0. That is, it determines \(x\) such that

$$p = \frac{1}{{\Gamma \left( a \right)}}\int_o^x {e^{ - t/b} } dt$$

where \(\Gamma(\cdot)\) is the gamma function. The probability that the random variable takes a value less than or equal to x is p. See the documentation for routine gamma for further discussion of the gamma distribution.

exponential uses bisection and modified regula falsi to invert the distribution function, which is evaluated using method gamma.

Parameters:

p - a double, the probability at which the function is to be evaluated

scale - a double, the scale parameter

Returns:

a double, the pth quantile x given the parameter values

See Also:

• Example

normal

public static double normal(double p)

Returns the inverse of the normal (Gaussian) cumulative probability distribution function.

Method normal evaluates the inverse of the distribution function \(\Phi\) of a standard normal (Gaussian) random variable, that is, normal \(({\rm p})=\Phi^{-1}(p)\), where

$$\Phi \left( x \right) = \frac{1}{{\sqrt {2\pi } }}\int_{ - \infty }^x {e^{ - t^2 /2} dt}$$

The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x. The standard normal distribution has a mean of 0 and a variance of 1.

Parameters:

p - a double, the probability at which the function is to be evaluated

Returns:

a double, the pth quantile x given the parameter values

See Also:

• Example

chi

public static double chi(double p,
 double df)

Returns the inverse of the chi-squared cumulative probability distribution function.

Method chi evaluates the inverse distribution function of a chi-squared random variable with df degrees of freedom, that is, with P = p and v = df, it determines x (equal to chi(p, df)), such that

$$P = \frac{1}{{2^{\nu /2} \Gamma \left( {\nu /2} \right)}}\int_0^x {e^{ - t/2} t^{\nu /2 - 1} } dt$$

where \(\Gamma(\cdot)\) is the gamma function. The probability that the random variable takes a value less than or equal to x is P.

For \(v \lt 40\), chi uses bisection, if \(v \ge 2\) or \(P \gt 0.98\), or regula falsi to find the point at which the chi-squared distribution function is equal to P. The distribution function is evaluated using chi.

For \(40 \le v \lt 100\), a modified Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.18) to the normal distribution is used, and normal is used to evaluate the inverse of the normal distribution function. For \(v \ge 100\), the ordinary Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.17) is used.

Parameters:

p - a double, the probability at which the function is to be evaluated

df - a double, the number of degrees of freedom. This must be at least 0.5.

Returns:

a double, the pth quantile x given the parameter values

See Also:

• Example

noncentralF

public static double noncentralF(double p,
 double dfn,
 double dfd,
 double lambda)

Returns the inverse of the noncentral F cumulative distribution function (CDF).

If \(X\) is a noncentral chi-square random variable with noncentrality parameter \(\lambda\) and \(\nu_1\) degrees of freedom, and \(Y\) is a chi-square random variable with \(\nu_2\) degrees of freedom which is statistically independent of \(X\), then $$F \;\; = \;\; (X/\nu_1)/(Y/\nu_2)$$ is a noncentral F-distributed random variable whose CDF is given by: $$CDF(f, \nu_1, \nu_2, \lambda) \;\; = \;\; \int_0^f {PDF(x, \nu_1, \nu_2, \lambda)dx} $$ where the probability density function \(PDF(x, \nu_1, \nu_2, \lambda)\) is given by: $$PDF(x, \nu_1, \nu_2, \lambda) \;\; = \;\; \Psi \; \sum_{k = 0}^\infty {\Phi_k} $$ $$\Psi \;\; = \;\; \frac{ e^{-\lambda/2}(\nu_1 x)^{\nu_1/2}(\nu_2)^{\nu_2/2} } { x \; (\nu_1 x \; + \; \nu_2)^{(\nu_1 + \nu_2)/2} \; \Gamma(\nu_2/2) }$$ $$\Phi_k \;\; = \;\; \frac{ R^k \; \Gamma(\frac{\nu_1 + \nu_2}{2} \; + \; k) } { k! \; \Gamma(\frac{\nu_1}{2} \; + \; k) } $$ $$R \;\; = \;\; \frac{ \lambda \nu_1 x }{ 2 (\nu_1 x \; + \; \nu_2)} $$ where \(\Gamma (\cdot)\) is the gamma function, \(\nu_1\) = dfn, \(\nu_2\) = dfd, \(\lambda\) = lambda, and \(p \;\; = \;\; CDF(f, \nu_1, \nu_2, \lambda)\) is the probability that \(F \le f\).

Method noncentralF evaluates $$f \;\; = \;\; CDF^{-1}(p, \nu_1, \nu_2, \lambda)$$ Method noncentralF uses bisection and modified regula falsi search algorithms to invert the distribution function \(CDF(f, \nu_1, \nu_2, \lambda)\), which is evaluated using method noncentralF. For sufficiently small p, an accurate approximation of \(CDF^{-1}(p, \nu_1, \nu_2, \lambda)\) can be used which requires no such inverse search algorithms.

Parameters:

p - a double scalar value representing the probability for which the inverse of the noncentral F cumulative distribution function is to be evaluated. p must be non-negative and less than one.

dfn - a double scalar value representing the number of numerator degrees of freedom. dfn must be positive.

dfd - a double scalar value representing the number of denominator degrees of freedom. dfd must be positive.

lambda - a double scalar value representing the noncentrality parameter. lambda must nonnegative.

Returns:

a double, the pth quantile x given the parameter values

noncentralBeta

public static double noncentralBeta(double p,
 double shape1,
 double shape2,
 double lambda)

Returns the inverse of the noncentral beta cumulative distribution function (CDF).

If \(Z\) is a noncentral chi-square random variable with noncentrality parameter \(\lambda\) and \(2 \alpha_1\) degrees of freedom, and \(Y\) is a chi-square random variable with \(2 \alpha_2\) degrees of freedom which is statistically independent of \(Z\), then

$$X \;\; = \;\; \frac{Z}{Z \; + \; Y} \;\; = \;\; \frac{\alpha_1 F}{\alpha_1 F \; + \; \alpha_2}$$

is a noncentral beta-distributed random variable and

$$F \;\; = \;\; \frac{\alpha_2 Z}{\alpha_1 Y} \;\; = \;\; \frac{\alpha_2 X}{\alpha_1 (1 \; - \; X)}$$

is a noncentral F-distributed random variable. The CDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F CDF:

$$CDF_{nc\beta}(x, \; \alpha_1, \; \alpha_2, \; \lambda) \;\; = \;\; CDF_{ncF}(f, \; 2 \alpha_1, \; 2 \alpha_2, \; \lambda)$$

where \(CDF_{nc\beta}(x, \; \alpha_1, \; \alpha_2, \; \lambda)\) is the noncentral beta CDF with \(x\) = x, \(\alpha_1\) = shape1, \(\alpha_2\) = shape2, and noncentrality parameter \(\lambda\) = lambda; \(CDF_{ncF}(f, \; 2 \alpha_1, \; 2 \alpha_2, \; \lambda)\) is the noncentral F CDF with argument f, numerator and denominator degrees of freedom \(2 \alpha_1\) and \(2 \alpha_2\) respectively, and noncentrality parameter \(\lambda\); and:

$$f \;\; = \;\; \frac{\alpha_2 x}{\alpha_1 (1 \; - \; x)}; \;\; x \;\; = \;\; \frac{\alpha_1 f}{\alpha_1 f \; + \; \alpha_2}$$

(See documentation for class Cdf method noncentralF for a discussion of how the noncentral F CDF is defined and calculated.)

Method noncentralBeta evaluates

$$x \;\; = \;\; CDF_{nc\beta}^{-1}(p, \; \alpha_1, \; \alpha_2, \; \lambda)$$

by first evaluating:

$$f \;\; = \;\; CDF_{ncF}^{-1}(p, \; 2 \alpha_1, \; 2 \alpha_2, \; \lambda)$$

and then solving for x using \(x \;\; = \;\; \frac{\alpha_1 f}{\alpha_1 f \; + \; \alpha_2}\). (See documentation for class Cdf method noncentralF for a discussion of how the inverse noncentral F CDF is calculated.)

Parameters:

p - a double scalar value representing the probability for which the inverse of the noncentral beta cumulative distribution function is to be evaluated. p must be non-negative and less than or equal to one.

shape1 - a double scalar value representing the first shape parameter. shape1 must be positive.

shape2 - a double scalar value representing the second shape parameter. shape2 must be positive.

lambda - a double scalar value representing the noncentrality parameter. lambda must be nonnegative.

Returns:

a double, the pth quantile x given the parameter values.

noncentralchi

public static double noncentralchi(double p,
 double df,
 double alam)

Returns the inverse of the noncentral chi-squared cumulative probability distribution function.

Method noncentralchi evaluates the inverse distribution function of a noncentral chi-squared random variable with df degrees of freedom and noncentrality parameter alam, that is, with P = p, \(\nu = {\rm df}\), and \(\lambda = {\rm alam}\), it determines \(c_{0} = \) noncentralchi(p, df, alam)), such that

$$P = \sum\limits_{i = 0}^\infty {\frac{e^{-\lambda/2}\left(\lambda/2\right)^i}{i!}} \int_0^{c_{0}} {\frac{x^{\left(\nu + 2i\right)/2-1}e^{ - x/2}} {2^{\left(\nu+2i\right)/2}{\Gamma\left(\frac{\nu+2i}{2}\right)}}} dx$$

where \(\Gamma (\cdot)\) is the gamma function. The probability that the random variable takes a value less than or equal to \(c_{0}\) is \(P\).

Method noncentralchi uses bisection and modified regula falsi to invert the distribution function, which is evaluated using noncentralchi. See noncentralchi for an alternative definition of the noncentral chi-squared random variable in terms of normal random variables.

Parameters:

p - a double, the probability at which the function is to be evaluated

df - a double, the number of degrees of freedom. This must be at least 0.5 but less than or equal to 200,000.

alam - a double, the noncentrality parameter. This must be nonnegative, and alam + df must be less than or equal to 200,000.

Returns:

a double, the pth quantile x given the parameter values

See Also:

• Example

studentsT

public static double studentsT(double p,
 double df)

Returns the inverse of the Student's t cumulative probability distribution function.

studentsT evaluates the inverse distribution function of a Student's t random variable with df degrees of freedom. Let v = df. If v equals 1 or 2, the inverse can be obtained in closed form, if v is between 1 and 2, the relationship of a t to a beta random variable is exploited and beta is used to evaluate the inverse; otherwise the algorithm of Hill (1970) is used. For small values of v greater than 2, Hill's algorithm inverts an integrated expansion in \(1/(1 + t^2/v)\) of the t density. For larger values, an asymptotic inverse Cornish-Fisher type expansion about normal deviates is used.

Parameters:

p - a double, the probability at which the function is to be evaluated

df - a double scalar value representing the number of degrees of freedom. This must be at least one.

Returns:

a double, the pth quantile x given the parameter values

See Also:

• Example

noncentralstudentsT

public static double noncentralstudentsT(double p,
 int idf,
 double delta)

Returns the inverse of the noncentral Student's t cumulative probability distribution function.

Method noncentralstudentsT evaluates the inverse distribution function of a noncentral t random variable with idf degrees of freedom and noncentrality parameter delta; that is, with \(\nu = idf\), \(\delta = delta\), it determines \(t_{0} = \) noncentralstudentsT(p, idf, delta)), such that

$$p = \int_{-{\infty}}^{t_{0}}{\frac{\nu^{\nu/2}e^{{-\delta^2}/2}} {{\sqrt{\pi}\Gamma\left(\nu/2\right)\left(\nu+x^2\right)}^{\left(\nu+1\right)/2}} } \sum\limits_{i = 0}^\infty {\Gamma\left(\left(\nu+i+1\right)/2\right)\left(\frac{\delta^i}{i!}\right) \left(\frac{2x^2}{\nu+x^2}\right)^{i/2} dx}$$

where \(\Gamma (\cdot)\) is the gamma function. The probability that the random variable takes a value less than or equal to \(t_{0}\) is P. See noncentralstudentsT for an alternative definition in terms of normal and chi-squared random variables. The method noncentralstudentsT uses bisection and modified regula falsi to invert the distribution function, which is evaluated using noncentralstudentsT.

Parameters:

p - a double, the probability at which the function is to be evaluated

idf - an int, the number of degrees of freedom. This must be positive.

delta - a double, the noncentrality parameter

Returns:

a double, the pth quantile x given the parameter values

See Also:

• Example

Weibull

public static double Weibull(double p,
 double gamma,
 double alpha)

Returns the inverse of the Weibull cumulative probability distribution function.

Parameters:

p - a double, the probability at which the function is to be evaluated

gamma - a double, the shape parameter

alpha - a double, the scale parameter

Returns:

a double, the pth quantile x given the parameter values

logNormal

public static double logNormal(double p,
 double mu,
 double sigma)

Returns the inverse of the standard lognormal cumulative probability distribution function.

Parameters:

p - a double, the probability at which the function is to be evaluated

mu - a double, the location parameter

sigma - a double, the shape parameter. sigma must be a positive.

Returns:

a double, the pth quantile x given the parameter values

extremeValue

public static double extremeValue(double p,
 double mu,
 double beta)

Returns the inverse of the extreme value cumulative probability distribution function.

Parameters:

p - a double, the probability at which the function is to be evaluated

mu - a double, the location parameter

beta - a double, the scale parameter

Returns:

a double, the pth quantile x given the parameter values

Rayleigh

public static double Rayleigh(double p,
 double alpha)

Returns the inverse of the Rayleigh cumulative probability distribution function.

Parameters:

p - a double, the probability at which the function is to be evaluated

alpha - a double, the scale parameter

Returns:

a double, the pth quantile x given the parameter values

uniform

public static double uniform(double p,
 double aa,
 double bb)

Deprecated.

use continuousUniform(double, double, double) instead

Returns the inverse of the uniform cumulative probability distribution function.

Parameters:

p - a double, the probability at which the function is to be evaluated

aa - a double, the minimum value

bb - a double, the maximum value

Returns:

a double, the pth quantile x given the parameter values

continuousUniform

public static double continuousUniform(double p,
 double a,
 double b)

Returns the inverse of the continuous uniform cumulative distribution function.

Parameters:

p - a double, the probability at which to evaluate the inverse cdf

a - a double, the lower parameter \(a\)

b - a double, the upper parameter \(b\)

Returns:

a double, the pth quantile x given the parameter values

discreteUniform

public static int discreteUniform(double p,
 int n)

Returns the inverse of the discrete uniform cumulative probability distribution function.

Parameters:

p - a double, the probability at which the function is to be evaluated

n - an int, the upper limit of the discrete uniform distribution

Returns:

a double, the pth quantile x given the parameter values

geometric

public static double geometric(double r,
 double pin)

Returns the inverse of the discrete geometric cumulative probability distribution function.

Parameters:

r - a double, the probability for which the inverse geometric function is to be evaluated

pin - a double, the probability parameter for each independent trial (the probability of success for each independent trial)

Returns:

a double, the pth quantile x given the parameter values

logistic

public static double logistic(double p,
 double mu,
 double sigma)

Returns the inverse of the logistic cumulative probability distribution function.

Parameters:

p - a double, the probability at which the function is to be evaluated

mu - a double, the location parameter

sigma - a double, the scale parameter

Returns:

a double, the pth quantile x given the parameter values

Pareto

public static double Pareto(double p,
 double xm,
 double k)

Returns the inverse of the Pareto cumulative probability density function.

Parameters:

p - a double, the probability at which the function is to be evaluated

xm - a double,the scale parameter

k - a double, the shape parameter

Returns:

a double, the pth quantile x given the parameter values