Expert Functions

BinomialExpert(n, p)

PMF:

\[P(X = k) = {n \choose k}p^k(1-p)^{n-k}, \quad \text{ for } k = 0,1,2, \ldots, n.\]

See also: Binomial Distribution (Wikipedia)

NegativeBinomialExpert(n, p)

PMF:

\[P(X = k) = \frac{\Gamma(k+r)}{k! \Gamma(r)} p^r (1 - p)^k, \quad \text{for } k = 0,1,2,\ldots.\]

See also: Negative Binomial Distribution (Wolfram)

PoissonExpert(λ)

PMF:

\[P(X = k) = \frac{\lambda^k}{k!} e^{-\lambda}, \quad \text{ for } k = 0,1,2,\ldots.\]

See also: Poisson Distribution (Wikipedia)

GammaCountExpert(m, s)

PMF:

\[P(X = k) = G(m k, s T) - G(m (k+1), s T), \quad \text{ for } k = 0,1,2, \ldots, n.\]

with

\[G(m k, s T) = \frac{1}{\Gamma(mk)} \int^{sT}_{0} u^{mk - 1} e^{-u} du\]

See also: Gamma Count Distribution (Arxiv)

BurrExpert(k, c, λ)

PDF:

\[f(x; k, c, \lambda) = \frac{kc}{\lambda} \left( \frac{x}{\lambda} \right)^{c-1} \left( 1+ \left( \frac{x}{\lambda} \right)^{c} \right)^{-k-1}, \quad x \geq 0\]

See also: Burr Distribution (Mathworks, implemented in this package), Burr Distribution (Wikipedia, with λ = 1)

GammaExpert(k, θ)

PDF:

\[f(x; k, \theta) = \frac{x^{k-1} e^{-x/\theta}}{\Gamma(k) \theta^k}, \quad x > 0\]

See also: Gamma Distribution (Wikipedia), shape-scale parameterization

InverseGaussianExpert(μ, λ)

PDF:

\[f(x; \mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\left(\frac{-\lambda(x-\mu)^2}{2\mu^2x}\right), \quad x > 0\]

See also: Inverse Gaussian Distribution (Wikipedia)

LogNormalExpert(μ, σ)

PDF:

\[f(x; \mu, \sigma) = \frac{1}{x \sqrt{2 \pi \sigma^2}} \exp \left( - \frac{(\log(x) - \mu)^2}{2 \sigma^2} \right), \quad x > 0\]

See also: Lognormal Distribution (Wikipedia)

WeibullExpert(k, θ)

PDF:

\[f(x; k, \theta) = \frac{k}{\theta} \left( \frac{x}{\theta} \right)^{k-1} e^{-(x/\theta)^k}, \quad x \geq 0\]

See also: Weibull Distribution (Wikipedia)