Distributions

Normal

Parametrization

Category Value
Type univariate, continuous
Support \(x \in (-\infty,\infty)\)
Parameter \(\theta_1 = \mu \in (-\infty,\infty)\), location
\(\theta_2 = \sigma \in (0,\infty)\), scale
Mean \(\mu\)
Variance \(\sigma^2\)

Probability density function

(24)\[ f(x \mid \mu ,\sigma) = {\frac {1}{\sqrt {2\pi \sigma ^{2}}}} \cdot e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}} = \phi \left( {\frac{{x - \mu }}{\sigma }} \right)\]

Cumulative distribution function

(25)\[ F(x \mid \mu ,\sigma ) = \frac{1}{2} \cdot \left( {1 + erf\left( {\frac{{x - \mu }}{{\sigma \cdot \sqrt 2 }}} \right)} \right) = \Phi \left( {\frac{{x - \mu }}{\sigma }} \right)\]

Lognormal

Parametrization

Category Value
Type univariate, continuous
Support \(x \in (0,\infty)\)
Parameter \(\theta_1 = \mu \in (-\infty,\infty)\), shape
\(\theta_2 = \sigma \in (0,\infty)\), scale
Mean \({e^{\left( {\mu + \frac{{{\sigma ^2}}}{2}} \right)}}\)
Variance \(\left( {{e^{{\sigma ^2}}} - 1} \right) \cdot {e^{2\mu + {\sigma ^2}}}\)

Probability density function

(26)\[ f(x \mid \mu ,\sigma) = {\frac {1}{x}}\cdot {\frac {1}{\sigma {\sqrt {2\pi \,}}}} \cdot \exp \left(-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}\right)\]

Cumulative distribution function

(27)\[ F(x, \mid \mu ,\sigma ) = \frac{1}{2} \cdot \left( {1 + erf\left( {\frac{{\ln (x) - \mu }}{{\sigma \cdot \sqrt 2 }}} \right)} \right) = \Phi \left( {\frac{{\ln (x) - \mu }}{\sigma }} \right)\]

Gumbel-max

Parametrization

Category Value
Type univariate, continuous
Support \(x \in (-\infty,\infty)\)
Parameter \(\theta_1 = \mu \in (-\infty,\infty)\), location
\(\theta_2 = \beta \in (0,\infty)\), scale
Mean \(\mu + \beta \cdot 0.5772\)[*]
Variance \(\frac{{{\pi ^2}}}{6} \cdot {\beta ^2}\)
[*]Euler-Mascheroni constant.

Probability density function

(28)\[ f(x \mid \mu ,\beta ) = \frac{1}{\beta } \cdot \exp \left( { - \frac{{x - \mu }}{\beta } - \exp \left( { - \frac{{x - \mu }}{\beta }} \right)} \right)\]

Cumulative distribution function

(29)\[ F(x \mid \mu ,\beta ) = \exp \left( { - \exp \left( { - \frac{{x - \mu }}{\beta }} \right)} \right)\]

Gamma

Parametrization

Category Value
Type univariate, continuous
Support \(x \in (0,\infty )\)
Parameter \(\theta_1 = k \in (0,\infty )\), shape
\(\theta_2 = \theta \in (0,\infty )\), scale
Mean \(k \cdot \theta\)
Variance \(k \cdot \theta^2\)

Probability density function

(30)\[ f(x \mid k,\theta ) = \frac{1}{{\Gamma(k) \cdot {\theta ^k}}} \cdot {x^{k - 1}} \cdot {e^{ - \frac{x}{\theta }}}\]

Where \(\Gamma(.)\) is the gamma function.

Cumulative distribution function

(31)\[ F(x \mid k,\theta ) = \frac{1}{{\Gamma (k)}} \cdot \gamma \left( {k,\frac{x}{\theta }} \right)\]

Where \(\gamma(.)\) is the lower incomplete gamma function.

Uniform

Parametrization

Category Value
Type univariate, continuous
Support \(x \in [a,b]\)
Parameter \(\theta_1 = a \in (-\infty,b)\), lower bound
\(\theta_2 = b \in (a,\infty)\), upper bound
Mean \({\tfrac {1}{2}} \cdot (a+b)\)
Variance \({\tfrac {1}{12}} \cdot (b-a)^{2}\)

Probability density function

(32)\[\begin{split} f(x \mid a,b ) = {\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}\end{split}\]

Cumulative distribution function

(33)\[\begin{split} F(x \mid a,b ) = {\begin{cases}0&{\text{for }}x<a\\{\frac {x-a}{b-a}}&{\text{for }}x\in [a,b)\\1&{\text{for }}x\geq b\end{cases}}\end{split}\]

Truncated Normal

Parametrization

Category Value
Type univariate, continuous
Support \(x \in [a,b]\)
Parameter \(\theta_1 = \mu \in (-\infty,\infty)\), location
  \(\theta_2 = \sigma \in (0,\infty)\), scale [†]
  \(\theta_3 = a \in (-\infty,b)\), lower bound
  \(\theta_4 = b \in (a,\infty)\), upper bound
Mean \(\mu +{\frac {\phi (\alpha )-\phi (\beta )}{Z}} \cdot \sigma\)
Variance \({\sigma ^2} \cdot \left[ {1 + \frac{{\alpha \cdot \phi (\alpha ) - \beta \cdot \phi (\beta )}}{Z} - {{\left( {\frac{{\phi (\alpha ) - \phi (\beta )}}{Z}} \right)}^2}} \right]\)
[†]For a more rigorous definition see the Wikipedia page of the truncated normal distribution.

Probability density function

(34)\[\begin{split} f(x \mid \mu ,\sigma ,a,b) = \left\{ {\begin{array}{*{20}{l}}{\frac{{\phi (\xi )}}{{\sigma \cdot Z}}{\mkern 1mu} }&{{\text{for }}x \in [a,b]}\\0&{{\rm{otherwise}}}\end{array}} \right.\end{split}\]

Where the auxiliary variables are defines as:

(35)\[\begin{split} \begin{array}{l}\xi = \frac{{x - \mu }}{\sigma },\;\alpha = \frac{{a - \mu }}{\sigma },\;\beta = \frac{{b - \mu}}{\sigma }\\Z = \Phi (\beta ) - \Phi (\alpha )\end{array}\end{split}\]

and \(\phi(.)\) and \(\Phi(.)\) are the standard normal probability density function (Eq. (24)) and cumulative distribution function (Eq. (25)), respectively.

Cumulative distribution function

(36)\[\begin{split} F(x\mid \mu ,\sigma ,a,b) = \left\{ {\begin{array}{*{20}{l}}0&{{\text{for }}x < a}\\{\frac{{\Phi (\xi ) - \Phi (\alpha )}}{Z}}&{{\text{for }}x \in [a,b)}\\1&{{\text{for }}x \ge b}\end{array}} \right.\end{split}\]

Exponential

Parametrization

Category Value
Type univariate, continuous
Support \(x \in (0,\infty )\)
Parameter \(\theta_1 = \lambda \in (0,\infty )\), rate
Mean \(\lambda^{-1}\)
Variance \(\lambda^{-2}\)

Probability density function

(37)\[ f(x \mid \lambda ) = \lambda \cdot {e^{ - \lambda x}}\]

Cumulative distribution function

(38)\[ F(x \mid \lambda ) = 1-e^{ - \lambda x}\]