Extra Discrete Distributions

AdditionalDistributions.BetaNegBinomial Type

BetaNegBinomial(r,α,β)

A Beta Negative Binomial is the compound distribution of the NegativeBinomial distribution where the probability of success p is distributed according to the Beta. It has three parameters: r, the number of successes number of successes until the experiment is stopped and two shape parameters $\alpha$, $\beta$

$$P(X=k)=\frac{B(r+k,\alpha +\beta )}{B(r,\alpha )k!}\frac{\mathrm{\Gamma }(k+\beta )}{\mathrm{\Gamma }(\beta )}$$

BetaNegBinomial()        # equivalent to BetaNegBinomial(1, 1, 1)
BetaNegBinomial(r)       # equivalent to BetaNegBinomial(r, 1, 1)
BetaNegBinomial(r, α)    # equivalent to BetaNegBinomial(r, α, α)

params(d)        # Get the parameters, i.e. (r , α, β)
succprob(d)    # Get the number of successes, i.e. r

External links

• Beta Negative Binomial distribution on Wikipedia

source

AdditionalDistributions.Borel Type

Borel(a)

A Borel distribution is a discrete probability distribution often used in branching processes and queueing theory. The probability mass function (PMF) of the Borel distribution is given by:

$$P(X=k)=\frac{e^{-ak}(ak{)}^{k-1}}{k!},k\in \{1,2,3,\dots \}$$

Borel()        # equivalent to Borel(0)

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

External link:

• Borel distribution on Wikipedia

source

AdditionalDistributions.Conway Type

Conway(λ, ν)

A Conway–Maxwell–Poisson distribution, often used to model overdispersed and underdispersed count data, is defined by the following probability mass function (PMF):

$$P(X=x)=\frac{{\lambda }^x}{(x!{)}^{\nu }Z(\lambda ,\nu )},x\in \{0,1,2,\dots \}$$

where:

• $Z(\lambda ,\nu )=\sum _{j=0}^{\mathrm{\infty }}\frac{{\lambda }^j}{(j!{)}^{\nu }}$ is a normalization constant that ensures the sum of probabilities equals 1.

Conway()        # equivalent to Conway(1, 1)
Conway(λ)       # equivalent to Conway(λ, 1)
Conway(λ, ν)    # equivalent to Conway(λ, ν)

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

External link:

• Conway Maxwell Poisson distribution on Wikipedia

source

AdditionalDistributions.Delaporte Type

Delaporte(λ,α,β)

A Delaporte distribution is a discrete probability distribution that can be viewed as a compound distribution. It combines a Poisson distribution (with mean λ) and a Gamma distribution (with shape parameters α and β). The probability mass function (PMF) of the Delaporte distribution is given by:

$$P(X=k)=\sum _{i=0}^k\frac{\mathrm{\Gamma }(\alpha +i){\beta }^i{\lambda }^{k-i}{e}^{-\lambda }}{\mathrm{\Gamma }(\alpha )i!(k-i)!},k\in \{0,1,2,\dots \}$$

Delaporte()        # equivalent to Delaporte(1, 1, 1)
Delaporte(λ)       # equivalent to Delaporte(λ, 1, 1)
Delaporte(λ, α)    # equivalent to Delaporte(r, α, α)

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

External link:

• Delaporte distribution on Wikipedia

source

AdditionalDistributions.FlorySchulz Type

FlorySchulz(a)

A Flory-Schulz distribution, commonly used in polymer chemistry to describe the distribution of chain lengths, is defined by the following probability mass function (PMF):

$$P(X=k)=a^{2}k(1-a{)}^{k-1},k\in \{1,2,3,\dots \}$$

where:

FlorySchulz()        # equivalent to FlorySchulz(0.5)

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

External link:

• Flory Schulz distribution on Wikipedia

source

AdditionalDistributions.GaussKuzmin Type

GaussKuzmin()

A Gauss-Kuzmin distribution is a discrete probability distribution that arises as the limiting distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed on (0, 1). The probability mass function (PMF) of the Gauss-Kuzmin distribution is given by:

$$P(X=k)=-{\log }_2(1-\frac{1}{(1+k{)}^2}),k\in \{1,2,3,\dots \}$$

where:

GaussKuzmin()        # equivalent to GaussKuzmin()

params(d)        # Get the parameters, i.e. none

External link:

• Gauss Kuzmin distributions on Wikipedia

source

AdditionalDistributions.Logarithmic Type

Logarithmic(a)

A Logarithmic distribution, also known as the log-series distribution, is defined by the following probability mass function (PMF):

$$P(X=k)=-\frac{1}{\log (1-p)}\cdot \frac{p^k}{k},k\in \{1,2,3,\dots \}$$

where:

Logarithmic()        # equivalent to Logarithmic(0.5)

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

External link:

• Logarithmic distribution on Wikipedia

source

AdditionalDistributions.Rademacher Type

Rademacher()

A Rademacher distribution is a discrete probability distribution where a random variate $X$ has a $50\mathrm{\%}$ chance of being $+1$ and a $50\mathrm{\%}$ chance of being $-1$.

$$P(X=k)=\{\begin{array}{ll}0.5& \text{for }k=-1,\\ 0.5& \text{for }k=+1.\end{array}$$

Rademacher()    # Rademacher distribution

External link:

• Rademacher distribution on Wikipedia

source

AdditionalDistributions.Yule Type

Yule(a)

The Yule distribution is a discrete probability distribution defined by the following probability mass function (PMF):

$$P(X=k)=a\ast B(k,a+1)$$

where:

• $B(k,a+1)$ is a beta function

Yule()        # equivalent to Yule(1)

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

External link

• Yule distribution on Wikipedia

source

AdditionalDistributions.Zeta Type

Zeta(s)

The Zeta distribution is a discrete probability distribution defined by the following probability mass function (PMF):

$$P(X=k)=\frac{1}{\zeta (s)}\cdot \frac{1}{k^s}$$

Where $\zeta$ is a Riemann Function

Zeta()      # equivalent to Zeta(1)

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

External link:

*Zeta Distribution on Wikipedia

source

AdditionalDistributions.ZIB Type

ZIB(n, θ, p)

The Zero-Inflated Binomial (ZIB) distribution is a discrete probability distribution that extends the binomial distribution by incorporating an excess of zero counts. The probability mass function (PMF) is defined as:

$$P(X=k)=\{\begin{array}{ll}\theta +(1-\theta )\cdot (1-p{)}^n& \text{if }k=0,\\ (1-\theta )\cdot (\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}& \text{if }k>0.\end{array}$$

ZIB()       # equivalent to ZIB(1, 0.5, 0.5)
ZIB(n)      # equivalent to ZIB(n, 0.5, 0.5)
ZIB(n, θ)   # equivalent to ZIB(n, θ, 0.5)

params(d)   # Get the parameters, i.e. (n, θ, p)

source

AdditionalDistributions.ZINB Type

ZINB(n, θ, p)

The Zero-Inflated Negative Binomial (ZINB) distribution is a discrete probability distribution that combines the negative binomial distribution with an excess of zeros. The probability mass function (PMF) is defined as:

$$P(X=k)=\{\begin{array}{ll}\theta +(1-\theta )\cdot (1-p{)}^r& \text{if }k=0,\\ (1-\theta )\cdot (\genfrac{}{}{0ex}{}{k+r-1}{k})p^r(1-p{)}^k& \text{if }k>0.\end{array}$$

ZINB()       # equivalent to ZINB(1, 0.5, 0.5)
ZINB(r)      # equivalent to ZINB(r, 0.5, 0.5)
ZINB(r, θ)   # equivalent to ZINB(r, θ, 0.5)

params(d)   # Get the parameters, i.e. (r, θ, p)

source

AdditionalDistributions.ZIP Type

ZIP(λ, p)

The Zero-Inflated Poisson (ZIP) distribution is a discrete probability distribution that models a scenario where there are more zeros in the data than would be expected from a standard Poisson distribution. It is defined by the following probability mass function (PMF):

$$P(X=k)=\{\begin{array}{ll}p+(1-p)\cdot e^{-\lambda }& \text{if }k=0,\\ (1-p)\cdot \frac{{\lambda }^k\cdot e^{-\lambda }}{k!}& \text{if }k\ge 1.\end{array}$$

julia ZIP() # equivalent to ZIP(1, 0.5) ZIP(λ) # equivalent to ZIP(λ, 0.5)

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

External link:

*Zero Inflated Poisson (ZIP) distribution on Wikipedia

source

AdditionalDistributions.Zipf Type

Zipf(N, s)

A Zipf distribution ...

$$P(X=k)=\frac{1}{H_{N,s}}\cdot \frac{1}{k^s}$$

Where $H_{N,s}$ is a generalized harmonic number

Zipf()      # equivalent to BetaNegBinomial(1, 1)
Zipf(N)     # equivalent to BetaNegBinomial(N, 1)

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

External link:

*Zipf distribution on Wikipedia

source

Index