StableDistributionS1 (FsODE)

FsODE

Documentation for 0.0.2

StableDistributionS1 Type

Namespace: Altaxo.Calc.Probability
Assembly: AltaxoCore.Redist.dll
Base Type: StableDistributionBase

Represents a stable distribution in Nolan's S1 parametrization.

The characteristic function in Nolan's S1 parametrization is:

            log(phi(t))= -scale^alpha |t|^alpha (1-i beta Sign(t) Tan(pi alpha/2)) + i location t    (for alpha not equal to 1)

and

            log(phi(t)) = -scale |t| (1+i beta Sign(t) (2/pi) Log(|t|)) + i location t  (for alpha equal to 1)

Reference: J.P.Nolan, Numerical calculation of stable densities and distribution functions. Communication is statistics - Stochastic models, 13, 759-774, 1999

Reference: S.Borak, W.Härdle, R.Weron, Stable distributions. SFB 649 Discussion paper 2005-2008, http://sfb649.wiwi.hu-berlin.de, ISSN 1860-5664

If you are interested in accurate calculations when beta is close to 1 or -1, you should use those functions which allow you to provide the parameter abe. This helps specifying beta with higher accuracy close to +1 or -1. For instance, by using abe=1E-30 and beta=1, it is possible to specify beta=1-1E-30, which is impossible otherwise since with the 64-bit representation of numbers.

Constructors

Constructor	Description
`StableDistributionS1()` Full Usage: `StableDistributionS1()`	Creates a new instance of the stable distribution in Nolan's parametrization with default parameters (alpha=1, beta=0) and the default generator.
`StableDistributionS1(generator)` Full Usage: `StableDistributionS1(generator)` Parameters: generator : `Generator` - Random number generator to be used with this distribution.	Creates a new instance of he stable distribution in Nolan's parametrization with default parameters (alpha=1, beta=0). generator : `Generator` Random number generator to be used with this distribution.
`StableDistributionS1(alpha, beta)` Full Usage: `StableDistributionS1(alpha, beta)` Parameters: alpha : `float` - Characteristic exponent of the distribution. beta : `float` - Skewness parameter of the distribution, in the range [-1,1].	Creates a new instance of he stable distribution in Nolan's parametrization with given parameters (alpha, beta) and the default random number generator. alpha : `float` Characteristic exponent of the distribution. beta : `float` Skewness parameter of the distribution, in the range [-1,1].
`StableDistributionS1(alpha, beta, abe)` Full Usage: `StableDistributionS1(alpha, beta, abe)` Parameters: alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0.	Creates a new instance of he stable distribution in Nolan's parametrization with given parameters (alpha, beta, abe) and the default random number generator. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0.
`StableDistributionS1(alpha, beta, scale, location)` Full Usage: `StableDistributionS1(alpha, beta, scale, location)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). scale : `float` - Scaling parameter (broadness of the distribution). location : `float` - Location parameter of the distribution.	Creates a new instance of he stable distribution in Nolan's parametrization with given parameters (alpha, beta, scale, location) and the default random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). scale : `float` Scaling parameter (broadness of the distribution). location : `float` Location parameter of the distribution.
`StableDistributionS1(alpha, beta, scale, location, generator)` Full Usage: `StableDistributionS1(alpha, beta, scale, location, generator)` Parameters: alpha : `float` - Characteristic exponent of the distribution. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. generator : `Generator` - Random number generator to be used with this distribution.	Creates a new instance of he stable distribution in Nolan's parametrization with given parameters (alpha, beta, scale, location) and the provided random number generator. alpha : `float` Characteristic exponent of the distribution. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. generator : `Generator` Random number generator to be used with this distribution.
`StableDistributionS1(alpha, beta, abe, scale, location)` Full Usage: `StableDistributionS1(alpha, beta, abe, scale, location)` Parameters: alpha : `float` - Characteristic exponent of the distribution. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution.	Creates a new instance of the stable distribution in Nolan's parametrization with given parameters (alpha, beta, abe, scale, location) and the default random number generator. alpha : `float` Characteristic exponent of the distribution. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution.
`StableDistributionS1(alpha, beta, abe, scale, location, generator)` Full Usage: `StableDistributionS1(alpha, beta, abe, scale, location, generator)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. generator : `Generator` - Random number generator to be used with this distribution.	Creates a new instance of he stable distribution in Nolan's parametrization with given parameters (alpha, beta, abe, scale, location) and the provided random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. generator : `Generator` Random number generator to be used with this distribution.

Instance members

Instance member	Description
`this.CDF` Full Usage: `this.CDF` Parameters: x : `float` - Argument. Returns: `float` The probability that the random variable of this probability distribution will be found at a value less than or equal to x. Modifiers: abstract	Calculates the cumulative distribution function. x : `float` Argument. Returns: `float` The probability that the random variable of this probability distribution will be found at a value less than or equal to x.
`this.Initialize` Full Usage: `this.Initialize` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution.	Initializes this instance of the distribution with the distribution parameters. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution.
`this.Maximum` Full Usage: `this.Maximum` Returns: `float` Modifiers: abstract	Gets the maximum possible value of distributed random numbers. Returns: `float`
`this.Mean` Full Usage: `this.Mean` Returns: `float` Modifiers: abstract	Gets the mean of distributed random numbers. For alpha<=1, it is not defined. For alpha>1, it is the StableDistributionS1._location parameter. Returns: `float`
`this.Median` Full Usage: `this.Median` Returns: `float` Modifiers: abstract	Gets the median of distributed random numbers. If beta=0, it is StableDistributionS1._location. For beta!=0, it is also defined, but not analytically expressable, and is not calcuated here (TODO, please help!). Returns: `float`
`this.Minimum` Full Usage: `this.Minimum` Returns: `float` Modifiers: abstract	Gets the minimum possible value of distributed random numbers. Returns: `float`
`this.Mode` Full Usage: `this.Mode` Returns: `float[]` Modifiers: abstract	Gets the mode of distributed random numbers. Returns: `float[]`
`this.NextDouble` Full Usage: `this.NextDouble` Returns: `float` A distributed double-precision floating point number. Modifiers: abstract	Returns a distributed floating point random number. Returns: `float` A distributed double-precision floating point number.
`this.PDF` Full Usage: `this.PDF` Parameters: x : `float` - Argument. Returns: `float` The relative likelihood for the random variable to occur at the point x. Modifiers: abstract	Calculates the probability density function. x : `float` Argument. Returns: `float` The relative likelihood for the random variable to occur at the point x.
`this.Quantile` Full Usage: `this.Quantile` Parameters: p : `float` - The probability p. Returns: `float` The point x at which the cumulative distribution function M:CDF() of argument x is equal to p. Modifiers: abstract	Calculates the quantile of the distribution function. p : `float` The probability p. Returns: `float` The point x at which the cumulative distribution function M:CDF() of argument x is equal to p.
`this.Variance` Full Usage: `this.Variance` Returns: `float` Modifiers: abstract	Gets the variance of distributed random numbers. Is finite only for alpha=2. Returns: `float`

Static members

Static member	Description
`StableDistributionS1.CDF(x, alpha, beta)` Full Usage: `StableDistributionS1.CDF(x, alpha, beta)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. Returns: `float` Cumulative distribution value at x.	Calculates the cumulative distribution function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. Returns: `float` Cumulative distribution value at x.
`StableDistributionS1.CDF(x, alpha, beta, scale, location)` Full Usage: `StableDistributionS1.CDF(x, alpha, beta, scale, location)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. Returns: `float` Cumulative distribution value at x.	Calculates the cumulative distribution function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. Returns: `float` Cumulative distribution value at x.
`StableDistributionS1.CDF(x, alpha, beta, tempStorage, precision)` Full Usage: `StableDistributionS1.CDF(x, alpha, beta, tempStorage, precision)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. tempStorage : `byref<obj>` - Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` - The required relative precision of calculation. Returns: `float` Cumulative distribution value at x.	Calculates the cumulative distribution function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. tempStorage : `byref<obj>` Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` The required relative precision of calculation. Returns: `float` Cumulative distribution value at x.
`StableDistributionS1.CDF(x, alpha, beta, scale, location, tempStorage, precision)` Full Usage: `StableDistributionS1.CDF(x, alpha, beta, scale, location, tempStorage, precision)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. tempStorage : `byref<obj>` - Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` - The required relative precision of calculation. Returns: `float` Cumulative distribution value at x.	Calculates the cumulative distribution function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. tempStorage : `byref<obj>` Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` The required relative precision of calculation. Returns: `float` Cumulative distribution value at x.
`StableDistributionS1.CDF(x, alpha, beta, abe)` Full Usage: `StableDistributionS1.CDF(x, alpha, beta, abe)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. Returns: `float` Cumulative distribution value at x.	Calculates the cumulative distribution function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. Returns: `float` Cumulative distribution value at x.
`StableDistributionS1.CDF(x, alpha, beta, abe, scale, location)` Full Usage: `StableDistributionS1.CDF(x, alpha, beta, abe, scale, location)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. Returns: `float` Cumulative distribution value at x.	Calculates the cumulative distribution function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. Returns: `float` Cumulative distribution value at x.
`StableDistributionS1.CDF(x, alpha, beta, abe, scale, location, tempStorage, precision)` Full Usage: `StableDistributionS1.CDF(x, alpha, beta, abe, scale, location, tempStorage, precision)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. tempStorage : `byref<obj>` - Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` - The required relative precision of calculation. Returns: `float` Cumulative distribution value at x.	Calculates the cumulative distribution function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. tempStorage : `byref<obj>` Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` The required relative precision of calculation. Returns: `float` Cumulative distribution value at x.
`StableDistributionS1.GetAgt1GnParameter(x, alpha, beta, abe, factorp, factorw, dev, logPrefactor)` Full Usage: `StableDistributionS1.GetAgt1GnParameter(x, alpha, beta, abe, factorp, factorw, dev, logPrefactor)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` factorw : `byref<float>` dev : `byref<float>` logPrefactor : `byref<float>`	x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` factorw : `byref<float>` dev : `byref<float>` logPrefactor : `byref<float>`
`StableDistributionS1.GetAlt1GnParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Full Usage: `StableDistributionS1.GetAlt1GnParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`	x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`
`StableDistributionS1.GetAlt1GpParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Full Usage: `StableDistributionS1.GetAlt1GpParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`	x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`
`StableDistributionS1.IsValidAlpha(alpha)` Full Usage: `StableDistributionS1.IsValidAlpha(alpha)` Parameters: alpha : `float` Returns: `bool`	alpha : `float` Returns: `bool`
`StableDistributionS1.IsValidBeta(beta)` Full Usage: `StableDistributionS1.IsValidBeta(beta)` Parameters: beta : `float` Returns: `bool`	beta : `float` Returns: `bool`
`StableDistributionS1.IsValidLocation(location)` Full Usage: `StableDistributionS1.IsValidLocation(location)` Parameters: location : `float` Returns: `bool`	location : `float` Returns: `bool`
`StableDistributionS1.IsValidScale(scale)` Full Usage: `StableDistributionS1.IsValidScale(scale)` Parameters: scale : `float` Returns: `bool`	scale : `float` Returns: `bool`
`StableDistributionS1.PDF(x, alpha, beta)` Full Usage: `StableDistributionS1.PDF(x, alpha, beta)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. Returns: `float` Probability density at x.	Calculates the probability density function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. Returns: `float` Probability density at x.
`StableDistributionS1.PDF(x, alpha, beta, scale, location)` Full Usage: `StableDistributionS1.PDF(x, alpha, beta, scale, location)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. Returns: `float` Probability density at x.	Calculates the probability density function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. Returns: `float` Probability density at x.
`StableDistributionS1.PDF(x, alpha, beta, scale, location, tempStorage, precision)` Full Usage: `StableDistributionS1.PDF(x, alpha, beta, scale, location, tempStorage, precision)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. tempStorage : `byref<obj>` - Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` - The required relative precision of calculation. Returns: `float` Probability density at x.	Calculates the probability density function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. tempStorage : `byref<obj>` Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` The required relative precision of calculation. Returns: `float` Probability density at x.
`StableDistributionS1.PDF(x, alpha, beta, abe, scale, location, tempStorage, precision)` Full Usage: `StableDistributionS1.PDF(x, alpha, beta, abe, scale, location, tempStorage, precision)` Parameters: x : `float` - Argument. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` - Scaling parameter (broadness) of the distribution, >0. location : `float` - Location parameter of the distribution. tempStorage : `byref<obj>` - Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` - The required relative precision of calculation. Returns: `float` Probability density at x.	Calculates the probability density function of the stable distribution in S1 parametrization. x : `float` Argument. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. scale : `float` Scaling parameter (broadness) of the distribution, >0. location : `float` Location parameter of the distribution. tempStorage : `byref<obj>` Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` The required relative precision of calculation. Returns: `float` Probability density at x.
`StableDistributionS1.Quantile(p, alpha, beta)` Full Usage: `StableDistributionS1.Quantile(p, alpha, beta)` Parameters: p : `float` - Probability p. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. Returns: `float` The value x, so that with a probability of p the random variable is	Calculates the quantile of the stable distribution in S1 parametrization. p : `float` Probability p. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. Returns: `float` The value x, so that with a probability of p the random variable is
`StableDistributionS1.Quantile(p, alpha, beta, abe)` Full Usage: `StableDistributionS1.Quantile(p, alpha, beta, abe)` Parameters: p : `float` - Probability p. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. Returns: `float` The value x, so that with a probability of p the random variable is	Calculates the quantile of the stable distribution in S1 parametrization. p : `float` Probability p. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. Returns: `float` The value x, so that with a probability of p the random variable is
`StableDistributionS1.Quantile(p, alpha, beta, abe, tempStorage, precision)` Full Usage: `StableDistributionS1.Quantile(p, alpha, beta, abe, tempStorage, precision)` Parameters: p : `float` - Probability p. alpha : `float` - Characteristic exponent of the distribution, in the range (0,2]. beta : `float` - Skewness parameter of the distribution, in the range [-1,1]. abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. tempStorage : `byref<obj>` - Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` - The required relative precision of calculation. Returns: `float` The value x, so that with a probability of p the random variable is	Calculates the quantile of the stable distribution in S1 parametrization. p : `float` Probability p. alpha : `float` Characteristic exponent of the distribution, in the range (0,2]. beta : `float` Skewness parameter of the distribution, in the range [-1,1]. abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is defined as 1-beta for beta>=0 or as 1+beta for beta<0. tempStorage : `byref<obj>` Temporary storage. For the first call, provide null as parameter. For subsequent calls, you can provide the object returned in the first call to speed up calculation. precision : `float` The required relative precision of calculation. Returns: `float` The value x, so that with a probability of p the random variable is