Extra Continuous Distributions

AdditionalDistributions.Alpha Type

Alpha(α, β)

An Alpha distribution is defined by the following probability density function (PDF):

$$f(x;\alpha ,\beta )=\frac{\beta }{\sqrt{2\pi }\mathrm{\Phi }(\alpha )x^2}\exp (-\frac{(\alpha -\frac{\beta }{x}{)}^2}{2}),x>0$$

where:

• $\alpha$ is a location parameter

• $\beta$ is a scale parameter

• $\mathrm{\Phi }$ is the cumulative distribution function (CDF) of the standard normal distribution.

Alpha()        # equivalent to Alpha(1, 1)

params(d)        # Get the parameters, i.e. (α, β)

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using Distributions, AdditionalDistributions

AdditionalDistributions.Argus Type

Argus(χ, c)

A Argus distribution, which is used in particle physics to describe the invariant mass of a decayed particle candidate. The probability density function (pdf) of the ARGUS distribution is given by:

$$f(x;\chi ,c)=\frac{{\xi }^3}{\sqrt{2\pi }\mathrm{\Phi }(\xi )}\cdot \frac{x}{c^2}\sqrt{1-\frac{x^2}{c^2}}\exp (-\frac{1}{2}{\xi }^2(1-\frac{x^2}{c^2}))$$

Argus()        # equivalent to Argus(1, 1)

params(d)        # Get the parameters, i.e. (χ, c)

External links:

• Argus distribution on Wikipedia

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AdditionalDistributions.Benini Type

A Benini distribution, which is a three-parameter distribution often used in economics and actuarial science. The probability density function (pdf) of the Benini distribution is given by:

$$f(x;\alpha ,\beta ,\sigma )=e^{-\alpha \log \left(\frac{x}{\sigma }\right)-\beta {[\log \left(\frac{x}{\sigma }\right)]}^2}\cdot (\frac{\alpha }{x}+\frac{2\beta \log \left(\frac{x}{\sigma }\right)}{x}),x>\sigma$$

Benini()        # equivalent to Benini(1, 1, 1)
Benini(α)        # equivalent to Benini(α, 1, 1)
Benini(α, β)        # equivalent to Benini(α, β, 1)

params(d)        # Get the parameters, i.e. (α, β, σ)

External links:

• Benini distribution on Wikipedia

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AdditionalDistributions.Benktander_Type1 Type

Benktander_Type1(a, b)

A Benktander_Type1 distribution, which is used in actuarial science for modeling heavy-tailed distributions. The probability density function (pdf) of the Benktander Type 1 distribution is given by:

$$f(x;a,b)=[(1+\frac{2b\log x}{a})(1+a+2b\log x)-\frac{2b}{a}]x^{-(2+a+b\log x)},x\ge 1$$

Benktander_Type1()        # equivalent to Benktander_Type1(1, 1)
Benktander_Type1(a)        # equivalent to Benktander_Type1(a, a(a+1)/2)

params(d)        # Get the parameters, i.e. (a, b)

External links:

• Benktander Type I distribution on Wikipedia

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AdditionalDistributions.Benktander_Type2 Type

Benktander_Type2(a, b)

A Benktander_Type2 distribution, which is used in actuarial science for modeling heavy-tailed distributions. The probability density function (pdf) of the Benktander Type 2 distribution is given by:

$$f(x;a,b)=\frac{e^{\frac{a}{b}(1-x^b)}\cdot x^{b-2}}{b}(a\cdot x^b-b+1),x\ge 1$$

Benktander_Type2()        # equivalent to Benktander_Type2(1, 1)
Benktander_Type2(a)        # equivalent to Benktander_Type2(a, 1)

params(d)        # Get the parameters, i.e. (a, b)

External links:

• Benktander Type II distribution on Wikipedia

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AdditionalDistributions.Bhattacharjee Type

Bhattacharjee(a, b, σ)

A Bhattacharjee distribution is a continuous univariate distribution where the mean follows a rectangular (uniform) distribution and the conditional distribution given the mean is normal. Specifically, if the mean θ follows a uniform distribution between a and b, and the conditional distribution of X given θ is normal with mean θ and standard deviation σ, then X follows a Bhattacharjee distribution.

$$f(x)=\frac{1}{b-a}[\mathrm{\Phi }\left(\frac{x-a}{\sigma }\right)-\mathrm{\Phi }\left(\frac{x-b}{\sigma }\right)]$$

Bhattacharjee()        # equivalent to Bhattacharjee(0, 1, 1)
Bhattacharjee(σ)       # equivalent to Bhattacharjee(0, 1, σ)

params(d)        # Get the parameters, i.e. (a , b, σ)

External link:

*Bhattacharjee Distributions on Technometrics

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AdditionalDistributions.BirnbaumSaunders Type

BirnbaumSaunders(μ, α, β)

A BirnbaumSaunders distribution, also known as the fatigue life distribution, is used to model life data and times to failure. It is defined by three parameters: the location parameter ( \mu ), the shape parameter ( \alpha ), and the scale parameter ( \beta ). The probability density function (PDF) of the Birnbaum-Saunders distribution is given by:

$$f(x)=\frac{\sqrt{\frac{x-\mu }{\beta }}+\sqrt{\frac{\beta }{x-\mu }}}{2\alpha (x-\mu )}\varphi \left(\frac{\sqrt{\frac{x-\mu }{\beta }}-\sqrt{\frac{\beta }{x-\mu }}}{\alpha }\right)$$

BirnbaumSaunders()        # equivalent to BirnbaumSaunders(0, 1, 1)
BirnbaumSaunders(σ)       # equivalent to BirnbaumSaunders(0, α, 1)
BirnbaumSaunders(σ,β)       # equivalent to BirnbaumSaunders(0, α, β)

params(d)        # Get the parameters, i.e. (μ , α, β)

External links:

• Birnbaum Saunders distribution on Wikipedia

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AdditionalDistributions.Bradford Type

Bradford(a)

A Bradford distribution is defined by the following probability density function (PDF):

$$f(x;a)=\frac{a}{\log (1+a)\cdot (1+a\cdot x)},0<x<1$$

where:

Bradford()        # equivalent to Bradford(1)

params(d)        # Get the parameters, i.e. a

External links:

• Bradford on Wikipedia

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AdditionalDistributions.Burr Type

Burr(c, k, λ)

An Burr distribution is defined by three parameters: $c$, $k$, and $\lambda$, where c and k are shape parameters and $\lambda$ is a scale parameter.

The probability density function (PDF) of the Burr distribution is given by:

$$f(x;c,k,\lambda )=\frac{ck}{\lambda }{\left(\frac{x}{\lambda }\right)}^{c-1}{(1+{\left(\frac{x}{\lambda }\right)}^c)}^{-(k+1)}$$

Burr()        # equivalent to Burr(1, 1, 1)
Burr(c)       # equivalent to Burr(c, 1, 1)
Burr(c, k)    # equivalent to Burr(c, k, 1)

params(d)        # Get the parameters, i.e. (c, k, λ)

External links:

• Burr distribution on Wikipedia

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AdditionalDistributions.CrystalBall Type

CrystalBall(α, m, x̄, σ)

A CrystalBall distribution is commonly used to model various lossy processes in high-energy physics. The probability density function (PDF) of the Crystal Ball distribution is defined as:

$$f(x;\alpha ,n,\overline{x},\sigma )=\{\begin{array}{lll}N\cdot \exp (-\frac{(x-\overline{x}{)}^2}{2{\sigma }^2}),& \text{for }\frac{x-\overline{x}}{\sigma }>-\alpha N\cdot A\cdot {(B-\frac{x-\overline{x}}{\sigma })}^{-n},& \text{for }\frac{x-\overline{x}}{\sigma }\le -\alpha \end{array}$$

where:

• $A={\left(\frac{n}{|\alpha |}\right)}^n\cdot \exp (-\frac{|\alpha {|}^2}{2})$

• $B=\frac{n}{|\alpha |}-|\alpha |$

• $N=\frac{1}{\sigma (C+D)}$

• $C=\frac{n}{|\alpha |}\cdot \frac{1}{n-1}\cdot \exp (-\frac{|\alpha {|}^2}{2})$

• $D=\sqrt{\frac{\pi }{2}}(1+\text{erf}\left(\frac{|\alpha |}{\sqrt{2}}\right))$

CrystalBall(α, m)        # equivalent to CrystalBall(α, m, 0, 1)
CrystalBall(α, m, x̄)     # equivalent to CrystalBall(α, m, x̄, 1)

params(d)        # Get the parameters, i.e. (α, m, x̄, σ)

External links:

• CrystalBall distribution on Wikipedia

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AdditionalDistributions.Dagum Type

Dagum(a, b, p)

A Dagum distribution is defined by three parameters: a, b, and p. It is commonly used in economics for modeling income distributions. The probability density function (PDF) of the Dagum distribution is given by:

$$f(x;a,b,p)=\frac{a\cdot p}{x}{\left(\frac{x}{b}\right)}^{ap}{({\left(\frac{x}{b}\right)}^a+1)}^{-(p+1)},x>0$$

Dagum()        # equivalent to Dagum(1, 1, 1)
Dagum(a)    # equivalent to Dagum(a, 1, 1)
Dagum(a, b)    # equivalent to Dagum(a, b, 1)

params(d)        # Get the parameters, i.e. (a, b, p)

External links:

• Dagum distribution on Wikipedia

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AdditionalDistributions.Gompertz Type

Gompertz(η , b)

A Gompertz distribution is commonly used to describe the distribution of adult lifespans and model survival data. The probability density function (PDF) of the Gompertz distribution is given by:

$$f(x;\eta ,b)=\eta b\exp (\eta +bx-\eta e^{bx}),x\ge 0$$

Gompertz()        # equivalent to Gompertz(1, 1)
Gompertz(η)        # equivalent to Gompertz(η, 1)

params(d)        # Get the parameters, i.e. (η, b)

External links:

• Gompertz distribution on Wikipedia

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AdditionalDistributions.Lomax Type

Lomax(α, λ)

A Lomax distribution, also known as the Pareto Type II distribution, is commonly used in business, economics, and actuarial science. The probability density function (PDF) of the Lomax distribution is given by:

$$f(x;\alpha ,\lambda )=\frac{\alpha }{\lambda }{(1+\frac{x}{\lambda })}^{-(\alpha +1)},x\ge 0$$

where:

• $\alpha$ is a shape parameter

• $\lambda$ is a scale parameter

Lomax()        # equivalent to Lomax(1, 1)
Lomax(α)       # equivalent to Lomax(α, 1)

params(d)      # Get the parameters, i.e. (α, λ)

External links:

• Lomax distribution on Wikipedia

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AdditionalDistributions.Maxwell Type

Maxwell(a)

A Maxwell-Boltzmann distribution, often used in physics to describe the distribution of particle speeds in idealized gases, is defined by the following probability density function (PDF):

$$f(x;a)=\sqrt{\frac{2}{\pi }}\frac{x^2}{a^3}\exp (-\frac{x^2}{2a^2}),x>0$$

Maxwell()        # equivalent to Maxwell(1)

params(d)        # Get the parameters, i.e. a

External links:

• Maxwell Boltzmann distribution on Wikipedia

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AdditionalDistributions.Nakagami Type

Nakagami(m, Ω)

A Nakagami distribution, often used to model physical phenomena in areas such as medical ultrasound imaging and communications, is defined by the following probability density function (PDF):

$$f(x;m,\mathrm{\Omega })=\frac{2m^m}{\mathrm{\Gamma }(m){\mathrm{\Omega }}^m}{x}^{2m-1}\exp (-\frac{m}{\mathrm{\Omega }}{x}^2),x\ge 0$$

where:

• m is a shape parameter

• $\mathrm{\Omega }$ is a spread parameter

Nakagami()        # equivalent to Nakagami(0.5, 1)
Nakagami(m)        # equivalent to Nakagami(m, 1)

params(d)        # Get the parameters, i.e. (m, Ω)

External links:

• Nakagami distribution on Wikipedia

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AdditionalDistributions.PERT Type

PERT(a, b, m)

A PERT distribution is commonly used in project management for modeling the uncertainty of task durations. The probability density function (PDF) of the PERT distribution is given by:

$$f(x;a,b,c)=\frac{(x-a{)}^{\alpha -1}(c-x{)}^{\beta -1}}{B(\alpha ,\beta )(c-a{)}^{\alpha +\beta -1}},a\le x\le c$$

where:

• $\alpha =1+4\frac{b-a}{c-a}$

• $\beta =1+4\frac{c-b}{c-a}$

• $B(\alpha ,\beta )$ is a beta function

PERT()        # equivalent to PERT(0, 0.5, 1)

params(d)        # Get the parameters, i.e. (a, b, m)

External links:

• PERT distribution on Wikipedia

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