probability distributions - nematriandiscrete (univariate) distributions the bernoulli distribution...
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Probability distributions
[Nematrian website page: ProbabilityDistributionsIntro, Β© Nematrian 2017] The Nematrian website contains information and analytics on a wide range of probability distributions, including: Discrete (univariate) distributions
- Bernoulli, see also binomial distribution - Binomial - Geometric, see also negative binomial distribution - Hypergeometric - Logarithmic - Negative binomial - Poisson - Uniform (discrete)
Continuous (univariate) distributions
- Beta - Beta prime - Burr - Cauchy - Chi-squared - Dagum - Degenerate - Error function - Exponential - F - Fatigue, also known as the Birnbaum-Saunders distribution - FrΓ©chet, see also generalised extreme value (GEV) distribution - Gamma - Generalised extreme value (GEV) - Generalised gamma - Generalised inverse Gaussian - Generalised Pareto (GDP) - Gumbel, see also generalised extreme value (GEV) distribution - Hyperbolic secant - Inverse gamma - Inverse Gaussian - Johnson SU - Kumaraswamy - Laplace - LΓ©vy - Logistic - Log-logistic - Lognormal - Nakagami - Non-central chi-squared - Non-central t
- Normal - Pareto - Power function - Rayleigh - Reciprocal - Rice - Studentβs t - Triangular - Uniform - Weibull, see also generalised extreme value (GEV) distribution
Continuous multivariate distributions
- Inverse Wishart Copulas (a copula is a special type of continuous multivariate distribution)
- Clayton - Comonotonicity - Countermonotonicity (only valid for π = 2, where π is the dimension of the input) - Frank - Generalised Clayton - Gumbel - Gaussian - Independence - t
The Nematrian website functions for fitting univariate distributions, creating random variates and calculating moments etc. now cover most of the above probability distributions, see ProbabilityDistributionsFunctions. In many cases these functions can cater for the traditional (textbook) forms of these distributions and variants that include additional shift and scale parameters.
location (i.e. shift) and scale variants [ProbabilityDistributionsIntro2] The location and scale of any probability distribution can be adjusted by using the (linear) transform π = π + βπ where π and β are constants (π adjusts the location, i.e. shifts the distribution, whilst β adjusts its scale). This leaves the skew and (excess kurtosis) unaltered but alters the mean and variance as πΈ(π) = π + βπΈ(π) and π£ππ(π) = β2π£ππ(π). In some cases the typical distributional specification already includes such components. For example, the normal distribution π(π, π2) is the location and scale adjusted version of the unit normal distribution π(0,1). In other cases the standard distributional specification does not include such adjustments. For example, the (standard) Studentβs t distribution depends one just one parameter, its degrees of freedom. The probability distribution orientated Nematrian web functions recognise location and/or scale adjusted variants of wide range of standard probability distributions.
Discrete (univariate) distributions
The Bernoulli distribution [BernoulliDistribution] A Bernoulli trial is an experiment that has one of two possible outcomes, βsuccessβ with probability π and βfailureβ with probability 1 β π. The Bernoulli distribution is a probability distribution that takes the value of 1 if such a trial is a βsuccessβ and 0 if it is a βfailureβ. The Bernoulli distribution is a special case of the binomial distribution, π΅(π, π), with π = 1. For characteristics of the Bernoulli distribution (e.g. mean, standard deviation etc.), please refer to the corresponding characteristics for the binomial distribution. For other probability distributions see here.
The binomial distribution [BinomialDistribution] The binomial distribution π΅(π, π) is the discrete probability distribution applicable to the number of successes in a sequence of π independent yes/no experiments each of which has a success probability of π. Each individual success/failure experiment is called a Bernoulli trial, so if π = 1 then the binomial distribution is a Bernoulli distribution. It has the following characteristics:
Distribution name Binomial distribution
Common notation π~π΅(π, π)
Parameters π = number of (independent) trials, positive integer π = probability of success in each trial, 0 β€ π β€ 1
Support π₯ β {0,1,β¦ , π} = ππ’ππππ ππ π π’ππππ π ππ
Probability mass function π(π₯) = (
ππ₯)ππ₯(1 β π)πβπ₯ =
π!
(π β π₯)! π₯!ππ₯(1 β π)πβπ₯
Cumulative distribution function πΉ(π₯) =β(
ππ)π
π(1 β π)πβππ₯
π=0
= πΌ1βπ(π β π₯, π₯ + 1)
Mean ππ Variance ππ(1 β π) Skewness 1 β 2π
βππ(1 β π)
(Excess) kurtosis 1 β 6π(1 β π)
ππ(1 β π)
Characteristic function (1 β π + ππππ‘)π
Other comments The Bernoulli distribution is π΅(1, π) and corresponds to the likelihood of success of a single experiment. Its probability mass function and cumulative distribution function are:
π(π₯) = πΉ(π₯) = {1 β π, π₯ = 0π, π₯ = 1
The Bernoulli distribution with π = 1 2β , i.e. π΅(1, 1 2β ), has the minimum possible excess kurtosis, i.e. β2.
The mode of π΅(π, π) is πππ‘((π + 1)π) if (π + 1)π is 0 or not an
integer and is π if (π + 1)π = π + 1. If (π + 1)π β {1,2,β¦ , π} then the distribution is bi-modal, with modes (π + 1)π and (π + 1)π β 1.
The binomial distribution is often used to model the number of successes in a sample size of π from a population size of π. Since such samples are not independent, the resulting distribution is actually a hypergeometric distribution and not a binomial distribution. However if π β« π then the binomial distribution becomes a good approximation to the relevant hypergeometric distribution and is thus often used.
In the above (ππ₯) is the binomial coefficient.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βbinomialβ. For details of other supported probability distributions see here.
The geometric distribution [GeometricDistribution] The geometric distribution describes the probability of π₯ successes in a sequence of independent experiments each with likelihood of success of π that arise before there is 1 failure. It is a special case of the negative binomial distribution. Note: different texts adopt slightly different definitions, e.g. it may be the total number of trials (i.e. 1 more than the above) in which case it may be called the shifted geometric distribution and/or π may denote the probability of failure rather than the probability of success.
The hypergeometric distribution [HypergeometricDistribution] The hypergeometric distribution describes the probability of π₯ successes in π draws from a finite population size π containing π successes without replacement. This contrasts with the binomial distribution which describes the probability of π₯ successes in π draws with replacement
Distribution name Hypergeometric distribution
Common notation π~π»π¦πππππππππ‘πππ(π,π, π)
Parameters π = population size, integral (π > 0) π = sample size, integral (0 < π β€ π)
π = number of tagged items, integral (0 < π β€ π)
Domain max(0, π + π βπ) β€ π₯ β€ max(π,π) , π₯ an integer
Probability mass function
π(π₯) =(ππ₯) (π βππ β π₯
)
(ππ)
Cumulative distribution function πΉ(π₯) = (π₯) =β
(ππ )(
π βππ β π
)
(ππ)
π₯
π=0
= 1 β(π
π + 1) (
π β ππ β π β 1
)
(ππ)
π
where
π = πΉ23 (1, π₯ + 1 βπ, π₯ + 1 β π; π₯ + 2,π + π₯ + 2 βπ β π; 1)
πΉππ is the generalised hypergeometric function, i.e.
πΉππ (π1, β¦ , ππ; π1, β¦ , ππ; π§) = β(π1)πβ¦(ππ)π(π1)πβ¦(ππ)π
π§π
π!
β
π=0
and (π)π involves the rising factorial or Pochhammer notation, i.e. (π)π = π(π + 1)(π + 2)β¦ (π + π β 1) and (π)0 = 1
Mean ππ
π
Variance ππ(π βπ)(π β π)
π2(π β 1)
Skewness (π β 2π)(π β 1)1 2β (π β 2π)
(ππ(π βπ)(π β π))1 2β(π β 2)
(Excess) kurtosis π΄ + π΅
πΆ
where
π΄ = (π β 1)π2(π(π + 1) β 6π(π βπ) β 6π(π βπ))
π΅ = 6ππ(π βπ)(π β π)(5π β 6) πΆ = ππ(π βπ)(π β π)(π β 2)(π β 3)
Characteristic function (π βππ
)
(ππ)
πΉ1(βπ,βπ; π βπ β π + 1; πππ‘)2
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βhypergeometricβ. For details of other supported probability distributions see here.
The logarithmic distribution [LogarithmicDistribution] The logarithmic distribution arises from following power series expansion:
β log(1 β π) = π +π2
2+π3
3+β―
This means that the function π(π₯) = βππ₯
π₯ log(1βπ), π₯ = 1,2,3, β¦ can naturally be interpreted as a
probability mass function since β π(π)βπ=1 = 1.
Distribution name Logarithmic distribution
Common notation π~πΏππ(π)
Parameters π = shape parameter (0 < π < 1)
Domain 1 β€ π₯ < +β, π₯ an integer
Probability mass function π(π₯) = β
ππ₯
π₯ log(1 β π)
Cumulative distribution function πΉ(π₯) = β
1
log(1 β π)β
ππ
π
π₯
π=1
= 1 +π΅π(π₯ + 1,0)
log(1 β π)
Mean βπ
(1 β π) log(1 β π)
Variance β
π(π + log(1 β π))
(1 β π)2(log(1 β π))2= π
Skewness β
π
(1 β π)3π3 2β log(1 β π)(1 + π +
3π
log(1 β π)+
2π2
log2(1 β π))
(Excess) kurtosis βπ
(1 β π)4π2 log(1 β π)π΄ β 3
where
π΄ = (1 + 4π + π2 +4π(1 + π)
log(1 β π)+
6π2
log2(1 β π)+
3π3
log3(1 β π))
Characteristic function log(1 β ππππ‘)
log(1 β π)
Other comments The logarithmic distribution has a mode of 1. If π is a random variable with Poission distribution and ππ, π = 1,β¦ is an infinite sequence of iid random variables each distributed πΏππ(π) then π =β ππππ=1 has a negative binomial distribution showing that the
negative binomial distribution is an example of a compound Poisson distribution
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βlogarithmicβ. For details of other supported probability distributions see here.
The negative binomial distribution [NegativeBinomialDistribution] The negative binomial distribution describes the probability of π₯ successes in a sequence of independent experiments each with likelihood of success of π that arise before there are π failures. In this interpretation π is a positive integer, but the distributional definition can also be extended to real values of π > 0. Note: different texts adopt slightly different definitions, e.g. with support starting at π₯ = π not π₯ = 0 and/or with π denoting probability of failure rather than probability of success.
Distribution name Negative binomial distribution
Common notation π~ππ΅(π, π) Parameters π = number of failures (π > 0)
π = probability of success in each experiment (0 < π < 1)
Support π₯ β {0,1,2,β¦ } Probability mass function π(π₯) = (
π₯ + π β 1π₯
)ππ₯(1 β π)π
If π is non-integral then is:
π(π₯) =Ξ(π₯ + π)
Ξ(π)Ξ(π₯ + 1)ππ₯(1 β π)π
Cumulative distribution function
πΉ(π₯) = 1 β πΌπ(π₯ + 1, π)
Mean ππ
1 β π
Variance ππ
(1 β π)2
Skewness 1 + π
βππ
(Excess) kurtosis 6
π+(1 β π)2
ππ
Characteristic function (1 β π
1 β ππππ‘)π
Other comments The geometric distribution is the same as the negative binomial distribution with parameter π = 1. Its pdf and cdf are therefore:
π(π₯) = π(1 β π)π₯ πΉ(π₯) = 1 β (1 β π)π₯+1
For the special case where π is an integer the negative binomial distribution is also called the Pascal distribution. The Poisson distribution is also a limiting case of the negative binomial:
ππππ π ππ(π) = limπββ
ππ΅ (π,π
π + π)
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βnegative binomialβ. For details of other supported probability distributions see here.
The Poisson distribution [PoissonDistribution] The Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time if the events occur with a known average rate and independently of the time since the last event.
Distribution name Poisson distribution
Common notation π~ππππ (π)
Parameters π = event rate (π > 0)
Support π₯ β {0,1,2,β¦ } Probability mass function
π(π₯) =ππ₯πβπ
π₯!
Cumulative distribution function πΉ(π₯) = πβπβ
ππ
π!
π₯
π=0
(can also be expressed using the incomplete gamma function)
Mean π Variance π
Skewness πβ1 2β (Excess) kurtosis πβ1 Characteristic function ππ(π
ππ‘β1) Other comments The median is approximately πππ‘(π + 1 3β β 0.02 πβ ).
The mode is πππ‘(π) if π is not integral. Otherwise the distribution is bi-modal with modes π and π β 1.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βpoissonβ. For details of other supported probability distributions see here.
The uniform (discrete) distribution [UniformDiscreteDistribution] The uniform (discrete) distribution involves equally probable outcomes that are spaced uniform intervals apart.
Distribution name Uniform (discrete) distribution
Common notation π~π(π, π, β) Parameters π = lower limit
π = upper limit (π < π) β = step size (π β π = (π β 1)β where π is a positive integer)
Support π₯ β {π, π + β,β¦ , π + (π β 1)β(= π)}
Probability mass function π(π₯) =
1
π πππ π₯ = π, π + β,β¦ , π + (π β 1)β
Cumulative distribution function
πΉ(π₯) = {
0 πππ π₯ < π
πππ‘ (π₯ β π
β+1
π) πππ π β€ π₯ < π
1 πππ π₯ β₯ π
Mean π + π
2
Variance (π β π)(π β π + 2β)
12
Skewness 0 (Excess) kurtosis
β6(π2 + 1)
5(π2 β 1)
Characteristic function 1
π(ππππ‘ β ππ(π+1)π‘
1 β πππ‘)
Other comments The median of this distribution is the same as its mean.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βuniform (discrete)β. For details of other supported probability distributions see here.
Continuous (univariate) distributions
The Beta distribution [BetaDistribution] The beta distribution describes a distribution in which outcomes are limited to a specific range, the probability density function within this range being characterised by two shape parameters.
Distribution name Beta distribution
Common notation π~π΅ππ‘π(πΌ, π½) Parameters πΌ = shape parameter (πΌ > 0)
π½ = shape parameter (π½ > 0)
Domain 0 β€ π₯ β€ 1
Probability density function π(π₯) =
π₯πΌβ1(1 β π₯)π½β1
π΅(πΌ, π½)
Cumulative distribution function
πΉ(π₯) =π΅π₯(πΌ, π½)
π΅(πΌ, π½)= πΌπ₯(πΌ, π½)
Mean πΌ
πΌ + π½
Variance πΌπ½
(πΌ + π½)2(πΌ + π½ + 1)
Skewness 2(π½ β πΌ)βπΌ + π½ + 1
(πΌ + π½ + 2)βπΌπ½
(Excess) kurtosis 6((πΌ β π½)2(πΌ + π½ + 1) β πΌπ½(πΌ + π½ + 2))
πΌπ½(πΌ + π½ + 2)(πΌ + π½ + 3)
Characteristic function
πΉ1 (πΌ; πΌ + π½; ππ‘)1 = 1 +β(βπΌ + π
πΌ + π½ + π
πβ1
π=0
)(ππ‘)π
π!
β
π=1
Other comments The beta distribution is also known as a beta distribution of the first
kind. Its mode is πΌβ1
πΌ+π½β2 for πΌ > 1, π½ > 1. There is no simple closed
form solution for its median. The beta distribution parameters are sometimes taken to include boundary parameters π, π (π < π) in which case its domain is π β€
π₯ β€ π, and its pdf and cdf are π(π₯) =(π₯βπ)πΌβ1(πβπ₯)π½β1
π΅(πΌ,π½) and πΉ(π₯) =
π΅π§(πΌ,π½)
π΅(πΌ,π½) where π§ =
π₯βπ
πβπ, its mean is
πΌπ+π½π
πΌ+π½, its mode is
(πΌβ1)π+(π½β1)π
πΌ+π½β2
for πΌ > 1, π½ > 1 and its variance is πΌπ½(πβπ)2
(πΌ+π½)2(πΌ+π½+1)
If π~Ξ(πΌ, π) and π~Ξ(π½, π) then π
π+π~π΅π(πΌ, π½). If π~π΅ππ‘π(πΌ, π½)
then π (1 β π)β ~π΅ππ‘ππππππ(πΌ, π½) and if π~π΅ππ‘π (π
2,π
2) then
ππ
π(1βπ)~πΉ(π,π) (if π > 0 and π > 0). The πβth order statistic of a
sample of size π from the uniform distribution has a beta distribution, π’(π)~π΅ππ‘π(π, π + 1 β π).
If π~π΅ππ‘π(1 + π (π β π) (π β π)β , 1 + π (π β π) (π β π)β ) then π + π(π β π)~ππππ‘(π, π,π, π), i.e. the Pert distribution is a special case of the beta distribution.
Its non-central moments are πΈ(ππ) = βπΌ+π
πΌ+π½+ππβ1π=0 .
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βbetaβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βbeta4β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Beta prime distribution [BetaPrimeDistribution]
Distribution name Beta prime distribution
Common notation π~π½β²(πΌ, π½) Parameters πΌ = shape parameter (πΌ > 0)
π½ = shape parameter (π½ > 0)
Domain π₯ β₯ 0
Probability density function π(π₯) =
π₯πΌβ1(1 + π₯)βπΌβπ½
π΅(πΌ, π½)
Cumulative distribution πΉ(π₯) = πΌπ₯ (1+π₯)β (πΌ, π½)
function
Mean πΌ
π½ β 1 ππ π½ > 1
Variance πΌ(πΌ + π½ β 1)
(π½ β 1)2(π½ β 2) ππ π½ > 2
Other comments The beta prime distribution is also called the inverted beta or the beta distribution of the second kind or the Pearson Type 6 distribution. The mode of the beta prime distribution is πΌβ1
πΌ+π½β2 for πΌ > 1, π½ > 1. There is no simple closed form expression
for its median. Its non-central moments (for integral π) are:
πΈ(ππ) =βπΌ + π β 1
π½ β π
π
π=1
=π΅(πΌ + π)π΅(π½ β π)
π΅(πΌ, π½)
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βbeta primeβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βbeta prime4β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Burr distribution [BurrDistribution]
Distribution name Burr
Common notation π~π΅π’ππ(πΌ, π½, π) or π~ππ(πΌ, π½, π)
Parameters π = shape parameter (π > 0) πΌ = shape parameter (πΌ > 0) π½ = shape parameter (π½ > 0)
Domain 0 β€ π₯ < +β Probability density function
π(π₯) =πΌπ½ππΌπ₯π½β1
(π + π₯π½)πΌ+1
Cumulative distribution function πΉ(π₯) = 1 β (
π
π + π₯π½)πΌ
Mean Ξ (πΌ β
1
π½)Ξ (1 +
1
π½)π1 π½β
Ξ(πΌ)
Variance
(
Ξ(πΌ β
2
π½)Ξ (1 +
2
π½) β
(Ξ(πΌ β1π½)Ξ (1 +
1π½))
Ξ(πΌ)
2
)
π2 π½β
Ξ(πΌ)
Other comments The Burr distribution is also known as the Burr Type XII distribution or the Singh-Maddala distribution (sometimes also called the generalised log-logistic distribution). Its non-central moments are:
πΈ(ππ) = Ξ (πΌ βπ
π½)Ξ (1 +
π
π½)ππ π½β
Ξ(πΌ) π = 1,2,β¦ , π < πΌπ½
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βburrβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βburr5β β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Cauchy distribution [CauchyDistribution]
Distribution name Cauchy distribution
Common notation π~πΆππ’πβπ¦(π, π) Parameters π = location parameter
π = scale parameter (π > 0)
Domain ββ < π₯ < +β
Probability density function π(π₯) = (ππ (1 + (
π₯ β π
π)2
))
β1
Cumulative distribution function
πΉ(π₯) =1
πarctan (
π₯ β π
π) +
1
2
Mean Does not exist
Variance Does not exist
Skewness Does not exist
(Excess) kurtosis Does not exist
Characteristic function exp(πππ‘ β π|π‘|) Other comments The quantile function of the Cauchy distribution is:
π(π) = π + π tan(π (π β1
2))
Its median is thus π. The Cauchy distribution is a special case of the stable (more precisely the sum stable) distribution family. The special case of the Cauchy distribution when π = 0 and π = 1 is called the standard Cauchy distribution. It coincides with the Studentβs t distribution with one degree of freedom. It has a
probability density function of π(π₯; 0,1) =1
π(1+π₯2).
If π~π(0,1) and π~π(0,1) are independent random variables then π
π~πΆππ’πβπ¦(0,1) and this can be used to generate random variates.
The Cauchy distribution is also known as the Cauchy-Lorentz or Lorentz distribution (especially amongst physicists).
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βcauchyβ. For details of other supported probability distributions see here.
The Chi-squared distribution [ChiSquaredDistribution] The chi-squared distribution with π degrees of freedom is the distribution of a sum of the squares of π independent standard normal random variables. A consequence is that the sum of independent chi-squared variables is also chi-squared distributed. It is widely used in hypothesis testing, goodness of fit analysis or in constructing confidence intervals. It is a special case of the gamma distribution.
Distribution name Chi-squared distribution
Common notation π~ππ2
Parameters π = degrees of freedom (positive integer)
Domain 0 β€ π₯ < +β Probability density function
π(π₯) =π₯π 2β β1 exp (β
π₯2)
2π 2β Ξ(π 2β )
Cumulative distribution function πΉ(π₯) =
Ξπ₯ 2β (π 2β )
Ξ(π 2β )
Mean π Variance 2π Skewness
2β2
π
(Excess) kurtosis 12
π
Characteristic function (1 β 2ππ‘)βπ 2β Other comments
Its median is approximately π (1 β2
9π)3
. Its mode is max(π β 2,0). Is
also known as the central chi-squared distribution (when there is a need to contrast it with the noncentral chi-squared distribution). In the special case of π = 2 the cumulative distribution function
simplifies to πΉ(π₯) = 1 β πβπ₯ 2β .
As π β β, (ππ2 β π) β2πβ β π(0,1) and ππΉ(π, π) β ππ
2
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βchi-squaredβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βchi-squared3β β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Dagum distribution [DagumDistribution]
Distribution name Dagum distribution
Common notation π~π·πππ’π(πΌ, π½, π)
Parameters π = shape parameter (π > 0) πΌ = shape parameter (πΌ > 0) π½ = scale parameter (π½ > 0)
Domain 0 β€ π₯ < +β Probability density function
π(π₯) =πΌπ (
π₯π½)πΌπβ1
π½ (1 + (π₯π½)πΌ
)π+1
Cumulative distribution function πΉ(π₯) =
π₯πΌπ
(π½πΌ + π₯πΌ)π
Mean
{βπ½
πΌ
Ξ (β1πΌ) Ξ (
1πΌ+ π)
Ξ(π)πΌ > 1
β otherwise
Variance
{β(π½
πΌ)2
(π΄ + π΅) πΌ > 2
β otherwise
where
π΄ =2πΌΞ (β
2πΌ) Ξ (
2πΌ+ π)
Ξ(π)
π΅ = (Ξ(β
1πΌ)Ξ (
1πΌ+ π)
Ξ(π))
2
Other comments Also known as the Dagum type 1 distribution. Is used in modelling income distributions. The cdf of a Dagum type 2 distribution adds a point mass at the origin and then follows a Dagum type 1 distribution over the positive halfline.
Its median is π½(21 πβ β 1)β1 πΌβ
and its mode is π½ (πΌπβ1
πΌ+1)1 πΌβ
.
Its non-central moments are:
πΈ(ππ) = Ξ (βπ
πΌ+ 1)Ξ (
π
πΌ+ π)
π½π
Ξ(π) π = 1,2,β¦ , π < πΌ
If π~π·πππ’π(πΌ, π½, π) then 1 πβ ~π΅π’ππ(πΌ, 1 π½β , π).
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βdagumβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βdagum4β β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Degenerate distribution [DegenerateDistribution]
The degenerate distribution characterises a distribution involving a single outcome. It can be viewed as the limiting case of many common distributions in which the scale parameter tends to zero, so the distribution function concentrates onto a single point. It is also called the Dirac delta function.
Distribution name Degenerate distribution (can also be viewed as a discrete distribution)
Common notation π~πΏπ Parameters π = location parameter
Domain π₯ = π
Probability density function π(π₯) = lim
πβ0(exp (β
1
2(π₯ β π
π)2
))
Cumulative distribution function
π(π₯) = {1, π₯ > π0, π₯ < 0
Mean π Variance 0 Skewness Does not exist
(Excess) kurtosis Does not exist
Characteristic function πππ‘π Nematrian web functions Given its degenerate form, no functions relating to the above distribution are accessible via the Nematrian web function library. For details of other supported probability distributions see here.
The error function distribution [ErrorFunctionDistribution] The error function distribution with parameter β is a special case of the normal distribution, i.e.
π (0,1
2β).
The exponential distribution [ExponentialDistribution] The exponential distribution describes the time between events if these events follow a Poisson process (i.e. a stochastic process in which events occur continuously and independently of one another). It is also called the negative exponential distribution. It is not the same as the exponential family of distributions.
Distribution name Exponential distribution
Common notation π~πΈπ₯π(π) Parameters π = inverse scale (i.e. rate) parameter (π > 0)
Domain 0 β€ π₯ < +β Probability density function π(π₯) = π exp(βππ₯)
Cumulative distribution function
πΉ(π₯) = 1 β exp(βππ₯)
Mean 1
π
Variance 1
π2
Skewness 2
(Excess) kurtosis 6
Characteristic function (1 β π π‘ πβ )β1 Other comments The exponential distribution is a special case of the Gamma
distribution, as if π~πΈπ₯π(π) then π~Ξ(1, 1 πβ ). The mode of an exponential distribution is 0. The quantile function, i.e. the inverse cumulative distribution function, is πΉβ1(π; π) =
βlog(1βπ)
π.
The non-central moments (π = 1,2,3,β¦ ) are πΈ(ππ) =Ξ(1+π)
ππ. Its
median is log 2
π.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βexponentialβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βexponential2β β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The F distribution [FDistribution]
The F distribution is also known as Snedecorβs F or the Fisher-Snedecor distribution. It commonly arises in statistical tests linked to analysis of variance.
Distribution name F distribution
Common notation π~πΉ(π1, π2)
Parameters π1 = degrees of freedom (first) (positive integer) π2 = degrees of freedom (second) (positive integer)
Domain 0 β€ π₯ < +β Probability density function
π(π₯) =1
π₯π΅(π1 2β , π2 2β )β
(π1π₯)π1π2
π2
(π1π₯ + π2)π1+π2
Cumulative distribution function πΉ(π₯) =
π΅π1π₯ (π1π₯+π2)β (π1 2β , π2 2β )
π΅(π1 2β , π2 2β )= πΌπ1π₯ (π1π₯+π2)β (π1 2β , π2 2β )
Mean π1π2 β 2
for π2 > 2
Variance 2π22(π1 + π2 β 2)
π1(π2 β 2)2(π2 β 4)
for π2 > 4
Skewness (2π1 + π2 β 2)β8(π2 β 4)
(π2 β 6)βπ1(π1 + π2 β 2) for π2 > 6
(Excess) kurtosis 12π1(5π2 β 22)(π1 + π2 β 2) + (π2 β 4)(π2 β 2)
2
π1(π2 β 6)(π2 β 8)(π1 + π2 β 2) for π2 > 8
Characteristic function Ξ (π1 + π22 )
Ξ (π22)
π (π12, 1 β
π22,βπ2π1ππ‘)
Where π(π, π, π§) is the confluent hypergeometric function of the second kind
Other comments The F distribution is a special case of the Pearson type 6 distribution. It is also a particular example of the beta prime distribution. If π1~π
2(π1) and π2~π2(π2) are independent random variables
then
π1 π1β
π2 π2β~πΉ(π1, π2)
Its mode is (π1β2)
π1
π2
π2+2 for π1 > 2. There is no simple closed form
for the median.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βfβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βf4β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Fatigue distribution [FatigueDistribution] The fatigue distribution, also known as the Birnbaum-Saunders distribution or the fatigue life distribution is used extensively to model failure times.
Distribution name (standard) Fatigue distribution
Common notation π~π΅ππππππ’π β πππ’πππππ (πΌ) Parameters πΌ = shape parameter (πΌ > 0)
Domain 0 < π₯ < +β Probability density function
π(π₯) =βπ₯ + β1 π₯β
2πΌπ₯π(
βπ₯ β β1 π₯β
πΌ)
where π(π₯) =1
β2πexp (β
π₯2
2)
Cumulative distribution function πΉ(π₯) = π(
βπ₯ β β1 π₯β
πΌ)
Mean 1 +
πΌ2
2
Variance πΌ2(4 + 5πΌ2)
4
Other comments Its quantile function, i.e. the inverse of its cdf, is:
πΉβ1(π) =1
4(πΌπβ1(π) + β4 + (πΌπβ1(π))
2)
2
Nematrian web functions
Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βfatigueβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βfatigue3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The FrechΓ©t distribution [FrechetDistribution] The FrechΓ©t distribution is a special case of the generalised extreme value (GEV) distribution. It characterises the distribution of βblock maximaβ under certain (relatively restrictive) conditions in situations where the tail index corresponds to fatter tails than arises with the normal distribution.
The gamma distribution [GammaDistribution] The gamma distribution is a two-parameter family of continuous probability distributions. Two different parameterisations are in common use, see below, with the (π, π) parameterisation being apparently somewhat more common in econometrics and the (π, π) parameterisation being somewhat more common in Bayesian statistics.
Distribution name Gamma distribution
Common notation π~Ξ(π, π) or Ξ(πΌ, π) Parameters Has two commonly used parameterisations:
π = shape parameter (π > 0) π = scale parameter (π > 0) or π = inverse scale (i.e. rate) parameter
(π > 0) where π = 1 πβ . Unless otherwise specified the material below assumes the first
parameterisation (i.e. using a scale parameter)
Domain 0 β€ π₯ < +β Probability density function
π(π₯) =π₯πβ1πβπ₯ πβ
ππΞ(π)
Cumulative distribution function πΉ(π₯) =
Ξπ₯ πβ (π)
Ξ(π)
Mean ππ
Variance ππ2 Skewness 2 βπβ (Excess) kurtosis 6 πβ
Moment generating function
(1 β ππ‘)βπ πππ π‘ < 1 πβ
Characteristic function (1 β πππ‘)βπ Other comments The gamma distribution can also be defined with a location
parameter, πΎ, say, in which case its domain is shifted to πΎ β€ π₯ <+β. Its mode is (π β 1)π for π β₯ 1. If π follows an exponential distribution with rate parameter π then π~Ξ(1, 1 πβ ). If π follows a chi-squared distribution, with π degrees of freedom, i.e. i.e. π~π2(π) then π~Ξ(π 2β , 2) and ππ~Ξ(π 2β , 2π). If π is integral then Ξ(π, π) is also called the Erlang distribution. It is the distribution of the sum of π independent exponential variables each with mean π = 1 πβ . Events that occur independently with some average rate are commonly modelled using a Poisson process. The waiting times between π occurrences of the event are then Erlang distributed whilst the number of events in a given amount of time is Poisson distributed. If π follows a Maxwell-Boltzmann distribution with parameter π then π2~ Ξ(3 2β , π = 2π2) . If π follows a skew logistic distribution with parameter π then log(1 + πβπ)~ Ξ(1, π). The gamma distribution is the conjugate prior for the precision (i.e. inverse variance) of a normal distribution and for the exponential distribution. The gamma distribution has the βsummationβ property that if ππ~Ξ(ππ, π) for π = 1,β¦ , π and the ππ are independent then β ππππ=1 ~Ξ(β ππ
ππ=1 , π).
Its non-central moments (π = 1,2,3, β¦ ) are πΈ(ππ) =Ξ(π+π)
Ξ(π)ππ.
There is in general no simple closed form for its median.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βgammaβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βgamma3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The generalised extreme value distribution [GEVDistribution] The generalised extreme value (or generalized extreme value) distribution characterises the behaviour of βblock maximaβ under certain (somewhat restrictive) regularity conditions. See also Nematrianβs webpages about Extreme Value Theory (EVT).
Distribution name Generalised extreme value (GEV) distribution (for maxima)
Common notation π~πΊπΈπ(π, π, π)
Parameters π = shape parameter π = location parameter π = scale parameter
Domain 1 + (π₯ β π
π) π > 0 π β 0
ββ < π₯ < β π = 0
Probability density function π(π₯) =
1
ππ(π₯)π+1πβπ(π₯)
where
π(π₯) =
{
(1 + π (
π₯ β π
π))
β1 πβ
π β 0
exp (βπ₯ β π
π) π = 0
Cumulative distribution function
πΉ(π₯) = πβπ(π₯)
Mean
{
π + πΞ(1 β π) β 1
π ππ π β 0, π < 1
π + ππΎ ππ π = 0β π β₯ 1
where πΎ is Eulerβs constant, i.e. limπββ
(β1
πβ logππ
π=1 )
Variance
{
π2
π2 β π12
π2 ππ π β 0, π < 1 2β
π2π2
6 ππ π = 0
β π β₯ 1 2β
Where ππ = Ξ(1 β ππ)
Skewness
{
π3 β 3π1π2 + 2π1
3
(π2 β π12)3 2β
ππ π β 0
12β6π(3)
π3 ππ π = 0
where π(π₯) is the Riemann zeta function, i.e. β1
ππ₯βπ=1 .
(Excess) kurtosis
{
π4 β 4π1π3 + 6π2π1
2 β 3π14
(π2 β π12)2
ππ π β 0
12
5 ππ π = 0
Other comments π defines the tail behaviour of the distribution. The sub-families defined by π = 0 (Type I), π > 0 (Type II) and π < 0 (Type III) correspond to the Gumbel, FrechΓ©t and Weibull families respectively. An important special case when analysing threshold exceedances involves π = 0 (and normally π > 0) and this special case may be referred to as πΊπΈπ(π, π).
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βgevβ. For details of other supported probability distributions see here.
The generalised gamma distribution [GeneralisedGammaDistribution] The generalised gamma (or generalized gamma) distribution is a generalisation of the gamma distribution that includes several of the parametric distributions typically used in survival analysis.
Distribution name Generalised gamma distribution
Common notation π~πΊππππππππ πππΊππππ(π, πΌ, π½) Parameters π = shape parameter (π > 0)
πΌ = shape parameter (πΌ > 0) π½ = scale parameter (π½ > 0)
Domain 0 β€ π₯ < +β
Probability density function π(π₯) =
π
π½Ξ(πΌ)(π₯
π½)ππΌβ1
exp(β(π₯
π½)π
)
Cumulative distribution function πΉ(π₯) =
Ξ(π₯ π½β )π(πΌ)
Ξ(πΌ)
Mean
π½Ξ (πΌ +
1π)
Ξ(πΌ)
Variance
π½2Ξ (πΌ +
2π)Ξ(πΌ) β Ξ (πΌ +
1π)2
Ξ(πΌ)2
Skewness Ξ (πΌ +
3π)Ξ(πΌ)2 β 3Ξ(πΌ +
2π)Ξ (πΌ +
1π)Ξ(πΌ) + 2Ξ (πΌ +
1π)3
(Ξ (πΌ +2π)Ξ(πΌ) β Ξ (πΌ +
1π)2
)
3 2β
(Excess) kurtosis π΄ + π΅ + πΆ + π·
(Ξ(πΌ +2π) Ξ(πΌ) β Ξ (πΌ +
1π)2
)
2 β 3
where:
π΄ = Ξ(πΌ +4
π)Ξ(πΌ)3
π΅ = β4Ξ(πΌ +3
π)Ξ (πΌ +
1
π)Ξ(πΌ)2
πΆ = 6Ξ(πΌ +2
π)Ξ (πΌ +
1
π)2
Ξ(πΌ)
π· = β3Ξ(πΌ +1
π)4
Other comments Its non-central moments are:
πΈ(ππ) =π½πΞ (πΌ +
ππ)
Ξ(πΌ)
The Weibull distribution is a special case with πΌ = 1. The gamma distribution is a special case with π = 1.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βgeneralised gammaβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βgeneralised gamma4β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The generalised inverse Gaussian distribution [GeneralisedInverseGaussianDistribution]
Distribution name Generalised inverse Gaussian distribution (GIG)
Common notation π~πΊπΌπΊ(π, π, π) Parameters π = parameter (π > 0)
π = parameter (π β₯ 0) π = parameter (π β₯ 0)
Domain 0 < π₯ < +β
Probability density function
π(π₯) =(βπ πβ )
π
2πΎπ(βππ)π₯πβ1 exp(β
1
2(π
π₯+ ππ₯))
where πΎπ(π₯) is the modified Bessel function of the third kind with index π.
Mean βππΎπ+1(βππ)
βππΎπ(βππ)
Variance
(π
π)(πΎπ+2(βππ)
πΎπ(βππ)β (
πΎπ+1(βππ)
πΎπ(βππ))
2
)
Characteristic function (
π
π β 2ππ‘)π 2β πΎπ(βπ(π β 2ππ‘))
πΎπ(βππ)
Other comments When π > 0 and π > 0 the non-central moments are:
πΈ(ππ) = (π
π)π 2β πΎπ+π(βππ)
πΎπ(βππ)
Some commentators use GIG to refer to the generalised integer gamma distribution (which is not the same as the generalised inverse Gaussian distribution).
Nematrian web functions This distribution is not currently supported within the Nematrian web function library. For details of other supported probability distributions see here.
The generalised Pareto distribution [GeneralisedParetoDistribution] The generalised Pareto distribution (generalized Pareto distribution) arises in Extreme Value Theory (EVT). If the relevant regularity conditions are satisfied then the tail of a distribution (above some suitably high threshold), i.e. the distribution of βthreshold exceedancesβ, tends to a generalized Pareto distribution. Care is needed with EVT because what we are in effect doing with it is to extrapolate into the tail of the distribution. Extrapolation is an intrinsically imprecise and subjective mathematical activity. We can in effect view the regularity conditions that need to be satisfied if EVT applies as corresponding to requiring that this extrapolation is done in a particular manner.
Distribution name Generalised Pareto distribution (GPD)
Common notation π~πΊππ·(π, π, π) Parameters π = shape parameter
π = location parameter π = scale parameter (π > 0)
Domain π β€ π₯ < +β π β₯ 0
π β€ π₯ β€ π βπ
ππ < 0
Probability density function
π(π₯) = {
1
π(1 + ππ§)β1β1 πβ π β 0
1
πexp(βπ§) π = 0
where
π§ =π₯ β π
π
Cumulative distribution function
πΉ(π₯) = {1 β (1 + ππ§)β1 πβ π β 0
1 β exp(βπ§) π = 0
Mean π +π
1 β π π < 1
Variance π2
(1 β 2π)(1 β π)2 π <
1
2
Skewness 2(1 + π)β1 β 2π
1 β 3π π <
1
3
(Excess) kurtosis 6(1 + π β 6π2 β 2π3)
(1 β 3π)(1 β 4π) π <
1
4
Other comments If π is uniformly distributed, π~π(0,1) then the variable π = π +
π(πβπ β 1) πβ ~πΊππ·(π, π, π).
The mean excess function for a GPD, i.e. π(π’) = πΈ(π β π’|π > π’) takes a particularly simple form which is linear in π, i.e.
π(π’) =π
1 β π+π(π’ β π)
1 β π
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βgeneralised paretoβ. For details of other supported probability distributions see here.
The Gumbel distribution [GumbelDistribution] The Gumbel distribution is a special case of the generalised extreme value distribution. It characterises the distribution of βblock maximaβ as per Extreme Value Theory (EVT) under certain (relatively restrictive) conditions.
The hyperbolic secant distribution
[Nematrian website page: HyperbolicSecantDistribution, Β© Nematrian 2015]
Distribution name Hyperbolic secant distribution
Common notation π~πππβ(π, π) Parameters π = scale parameter (π > 0)
π = location parameter
Domain ββ < π₯ < +β Probability density function
π(π₯) =sech (
π2 π§)
2π
where
π§ =π₯ β π
π
Cumulative distribution function
πΉ(π₯) =2
πarctan (exp (
π
2π§))
Mean π Variance π2 Skewness 0 (Excess) kurtosis 2
Characteristic function πβππ‘π sech(ππ‘) Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βhyperbolic secantβ. For details of other supported probability distributions see here.
The inverse gamma distribution [InverseGammaDistribution] The inverse gamma distribution describes the distribution of the reciprocal of a variable distributed according to the gamma distribution.
Distribution name Inverse gamma
Common notation
Parameters πΌ = shape parameter (πΌ > 0) π½ = scale parameter (π½ > 0)
Domain 0 < π₯ < +β Probability density function
π(π₯) =exp (β
π½π₯)
π½Ξ(πΌ)(π₯ π½β )πΌ+1
Cumulative distribution function πΉ(π₯) = 1 β
Ξπ½ π₯β (πΌ)
Ξ(πΌ)
Mean π½
πΌ β 1 πππ πΌ > 1
Variance π½2
(πΌ β 1)2(πΌ β 2) πππ πΌ > 2
Skewness 4βπΌ β 2
πΌ β 3 πππ πΌ > 3
(Excess) kurtosis 30πΌ β 66
(πΌ β 3)(πΌ β 4) πππ πΌ > 4
Characteristic function 2(βππ½π‘)πΌ 2β
Ξ(πΌ)πΎπΌ(2ββππ½π‘)
Other comments Also called the log Pearson type 5 distribution. Its mode is π½ (πΌ + 1)β
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βinverse gammaβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βinverse gamma3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The inverse Gaussian distribution [InverseGaussianDistribution] While the Gaussian (i.e. normal distribution describes a Brownian motionβs level at a fixed time, the inverse Gaussian distribution describes the distribution of time a Brownian motion starting at 0 with positive drift takes to reach a fixed positive level.
Distribution name Inverse Gaussian distribution
Common notation π~πΌπΊ(π, π) Parameters π = parameter (π > 0)
π = parameter (π > 0)
Domain 0 < π₯ < +β
Probability density function
π(π₯) = βπ
2ππ₯3exp (β
π(π₯ β π)2
2π2π₯)
Cumulative distribution function πΉ(π₯) = π(β
π
π₯(π₯
πβ 1)) + π(ββ
π
π₯(π₯
π+ 1)) exp (
2π
π)
Mean π Variance π3 πβ Skewness
3 (π
π)1 2β
(Excess) kurtosis 15π
π
Characteristic function
exp(π
π(1 β β1 β
2π2ππ‘
π))
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βinverse gaussianβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βinverse gaussian3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Johnson SU distribution [JohnsonSUDistribution]
Distribution name Johnson SU distribution
Common notation π~π½πβππ ππππ(πΎ, πΏ, π, π) or π~ππ(πΎ, πΏ, π, π)
Parameters πΎ = shape parameter πΏ = shape parameter (πΏ > 0) π = location parameter
π = scale parameter (π > 0)
Domain ββ < π₯ < +β
Probability density function
π(π₯) =πΏ exp (β
12(πΎ + πΏ sinhβ1 π§)2)
πβ2πβπ§2 + 1
where
π§ =π₯ β π
π
Note sinhβ1 π§ = log(π§ + βπ§2 + 1)
Cumulative distribution function
πΉ(π₯) = π(πΎ + πΏ sinhβ1 π§)
Mean π β πβπ€ sinhΞ© where π€ = exp(πΏβ2) and Ξ© = πΎ πΏβ
Variance π2
2(π€ β 1)(π€ cosh(2Ξ©) + 1)
Skewness βπ3βπ€(π€ β 1)2(π€(π€ + 2) sinh(3Ξ©) + 3 sinhΞ©)
4π3
where π = βπ2
2(π€ β 1)(π€ cosh(2Ξ©) + 1)
(Excess) kurtosis π4(π€ β 1)2(π΄ + π΅ + πΆ)
8π4β 3
where π΄ = π€2(π€4 + 2π€3 + 3π€2 β 3) cosh(4Ξ©)
π΅ = 4π€2(π€ + 2) cosh(2Ξ©) πΆ = 3(2π€ + 1)
Nematrian web functions
Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βjohnsonsuβ. For details of other supported probability distributions see here.
The Kumaraswamy distribution [KumaraswamyDistribution] The Kumaraswamy distribution describes a distribution in which outcomes are limited to a specific range, the probability density function within this range being characterised by two shape parameters. It is similar to the beta distribution but possibly easier to use because it has simpler analytical expressions for its probability density function and cumulative distribution function.
Distribution name Kumaraswamy distribution
Common notation π~πΎπ’πππππ π€πππ¦(πΌ, π½) Parameters πΌ = shape parameter (πΌ > 0)
π½ = shape parameter (π½ > 0)
Domain 0 β€ π₯ β€ 1 Probability density function π(π₯) = πΌπ½π₯πΌβ1(1 β π₯πΌ)π½β1 Cumulative distribution function
πΉ(π₯) = 1 β (1 β π₯πΌ)π½
Mean π½π΅ (1 +
1
πΌ, π½)
Variance (π½π΅ (1 +
2
πΌ, π½) β π½2π΅ (1 +
1
πΌ, π½)
2
)
Other comments A standard Kumaraswamy distribution has π = 0 and π = 1. Its non-central moments are given by:
πΈ(ππ) =π½Ξ(1 + π πΌβ )Ξ(π½)
Ξ(1 + π½ + π πΌβ )= π½π΅ (1 +
π
πΌ, π½)
and its median is (1 β 2β1 π½β )1 πΌβ
and its mode is (πΌβ1
πΌπ½β1)1 πΌβ
(for πΌ β₯
1,π½ β₯ 1, (πΌ, π½) β (1,1).
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βkumaraswamyβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a
DistributionName of βkumaraswamy4β β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Laplace distribution [LaplaceDistribution] The Laplace distribution is akin to two exponential distributions spliced together.
Distribution name Laplace distribution
Common notation π~πΏππππππ(π, π) Parameters π = location parameter
π = scale parameter (π > 0)
Domain ββ < π₯ < +β
Probability density function π(π₯) =
1
2πexp(β
|π₯ β π|
π)
Cumulative distribution function
πΉ(π₯) = {
1
2exp (
π₯ β π
π) π₯ β€ π
1 β1
2exp (β
π₯ β π
π) π₯ > π
Mean π
Variance 2π2 Skewness 0 (Excess) kurtosis 3
Characteristic function ππππ‘
1 + π2π‘2
Other comments The median and mode are π. The inverse cdf (i.e. quantile) function is
πΉβ1(π) = π β π sgn (π β1
2) log (1 β 2 |π β
1
2|).
If π~π (β1
2,1
2) and π = π β π sgn(π) log(1 β 2|π|) then
π~πΏππππππ(π, π). It is sometimes referred to as the double exponential distribution (but this term is also apparently also used sometimes of the Gumbel distribution)
Nematrian web functions
Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βlaplaceβ. For details of other supported probability distributions see here.
The LΓ©vy distribution [LevyDistribution] The time of hitting a single point π (different from the starting point of 0) of a Brownian motion without a mean drift follows the LΓ©vy distribution with π = π2. With a mean drift it follows an inverse Gaussian meaning that the LΓ©vy distribution is a special case of the inverse Gaussian distribution.
Distribution name LΓ©vy distribution
Common notation π~πΏππ£π¦(π)
Parameters π = scale parameter (π > 0)
Domain 0 < π₯ < +β
Probability density function
π(π₯) =1
πβ2π
exp (π2π₯)
(π₯ πβ )3 2β
Cumulative distribution function πΉ(π₯) = 2 β 2π (β
π
π₯)
Mean β
Variance β
Skewness undefined
(Excess) kurtosis undefined
Characteristic function πβββ2πππ‘ Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βlevyβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βlevy2β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The logistic distribution
[LogisticDistribution] The logistic distribution has a similar shape to the normal distribution but has heavier tails.
Distribution name Logistic distribution
Common notation π~πΏππππ π‘ππ(π, π )
Parameters π = location parameter π = scale parameter (π > 0)
Domain ββ < π₯ < +β Probability density function
π(π₯) =exp(βπ§)
π (1 + exp(βπ§))2
where
π§ =π₯ β π
π
Cumulative distribution function
πΉ(π₯) =1
1 + exp(βπ§)
Mean π Variance π2π 2
3
Skewness 0
(Excess) kurtosis 6
5
Characteristic function ππππ‘ππ π‘
sinh(ππ π‘)
Other comments The logistic distribution is sometimes called the sech-square(d) distribution because its pdf can also be expressed in terms of the hyperbolic secan function:
π(π₯) =exp(βπ§)
π(1 + exp(βπ§))2=1
4πsech2 (
π§
2)
It should not then be confused with the hyperbolic secant distribution.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βlogisticβ. For details of other supported probability distributions see here.
The log-logistic distribution [LogLogisticDistribution]
Distribution name Log-logistic distribution
Common notation π~πΏπΏ(πΌ, π½) Parameters πΌ = scale parameter (πΌ > 0)
π½ = shape parameter (π½ > 0)
Domain 0 β€ π₯ < +β
Probability density function
π(π₯) =(π½πΌ) (
π₯πΌ)
π½β1
(1 + (π₯πΌ)
π½)2
Cumulative distribution function
πΉ(π₯) =1
1 + (π₯πΌ)βπ½
Mean πΌπ csc(π) ππ π½ > 1 where π = π π½β and csc π₯ = 1 sin π₯β is the cosecant function
Variance πΌ2(2π csc(2π) β π2 csc2 π) ππ π½ > 2 Other comments Is also called the Fisk distribution. Its non-central moments are (if
π < π½):
πΈ(ππ) = πΌππ΅(1 β π π½β , 1 + π π½β ) =πΌπππ
sin(ππ)
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βloglogisticβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βloglogistic3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The lognormal distribution [LognormalDistribution]
Distribution name Lognormal distribution
Common notation π~ππππ(π, π2) Parameters π = scale parameter (π > 0)
π = location parameter
Domain 0 < π₯ < +β Probability density function
π(π₯) =
exp (β12 (log π₯ β π
π )2
)
π₯πβ2π
Cumulative distribution function
πΉ(π₯) = π (log π₯ β π
π) =
1
2+1
2πππ (
log π₯ β π
β2π2)
Mean ππ+π2 2β
Variance (π(π2) β 1)π2π+π
2
Skewness (π(π2) + 2)βπ(π
2) β 1
(Excess) kurtosis π(4π2) + 2π(3π
2) + 3π(2π2) β 6
Characteristic function No simple expression that is not divergent
Other comments The median of a lognormal distribution is ππ and its mode is ππβπ2.
The truncated moments of ππππ(π, π2) are:
β« π₯ππ(π₯)ππ₯
π
πΏ
= πππ+π2π2 2β (π (
logπ β π
πβ ππ)
β π (log πΏ β π
πβ ππ))
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βlognormalβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βlognormal4β β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Nakagami distribution [LognormalDistribution]
Distribution name Nakagami distribution
Common notation π~ππππππππ(π,Ο)
Parameters π = parameter (π β₯ 1 2β ) Ο = parameter (Ο > 0)
Domain 0 β€ π₯ < +β Probability density function
π(π₯) =2ππ
Ξ(π)Οππ₯2πβ1 exp (β
π
Οπ₯2)
Cumulative distribution function πΉ(π₯) =
Ξππ₯2 Οβ (π)
Ξ(π)
Mean Ξ (π +12)
Ξ(π)(π
π)1 2β
Variance
π(1 β1
π(Ξ(π +
12)
Ξ(π))
2
)
Other comments Its median is βπ and its mode is
1
β2((2πβ1)π
π)1 2β
.
If π~Ξ(π, π) then π = βπ~ππππππππ(π, ππ)
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βnakagamiβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βnakagami3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The non-central chi-squared distribution [NoncentralChiSquaredDistribution]
Distribution name Non-central chi-squared distribution
Common notation π~π2(π, π) Parameters π = degrees of freedom (positive integer)
π = non-centrality parameter (π β₯ 0)
Domain 0 β€ π₯ < +β
Probability density function
π(π₯) =π₯π 2β β1 exp (β
π₯2)
2π 2β Ξ(π 2β )πΉ10 (; π 2β ; ππ₯ 4β )
Cumulative distribution function
πΉ(π₯) = 1 β ππ 2β (βπ,βπ₯)
where ππ(π, π) is the Marcum-Q function.
Mean π + π Variance 2(π + 2π)
Skewness 2β2(π + 3π)
(π + 2π)3 2ββ8
π
(Excess) kurtosis 12(π + 4π)
(π + 2π)2
Characteristic function (1 β 2ππ‘)βπ 2β exp (β
πππ‘
1 β 2ππ‘)
Other comments The non-central chi-squared distribution with π degrees of freedom and non-centrality parameter π is the distribution of the sum of the squares of π independent normal distributions each with unit
standard deviation but with non-zero means ππ where π = β ππ2π
π=1 . The (standard) chi-squared distribution is a special case of it with π =0.
Nematrian web functions This distribution is not currently supported within the Nematrian web function library. For details of other supported probability distributions see here.
The non-central t distribution [LognormalDistribution]
Distribution name (Standard) non-central t distribution (NCT)
Common notation π~ππΆπ(π, π)
Parameters π = degrees of freedom (π > 0, usually π is an integer although in some situations a non-integral π can arise)
π = non-centrality parameter
Domain ββ < π₯ < +β
Probability density function
π(π₯) =
{
π
π₯(πΉπ+2,π (π₯β1 + 2 πβ ) β πΉπ,π(π₯)) ππ π₯ β 0
Ξ((π + 1) 2β )
βππΞ(π 2β )exp (β
π2
2) ππ π₯ = 0
where πΉπ,π(π₯) is the NCT cumulative distribution function
Cumulative distribution function πΉπ,π(π₯) = {
ππ,π(π₯) ππ π₯ β₯ 0
1 β ππ,βπ(βπ₯) ππ π₯ < 0
where
ππ,π(π₯) = π(βπ) +1
2β(πππΌπ¦ (π +
1
2,π
2) + πππΌπ¦ (π + 1,
π
2))
β
π=0
π¦ =π₯2
π₯2 + π
ππ =1
π!(π2
2)
π
exp(βπ2
2)
ππ =π
β2Ξ (π +32)(π2
2)
π
exp(βπ2
2)
Mean
πβπ
2
Ξ (π β 12 )
Ξ (π2)
ππ π > 1
Variance π(1 + π2)
π β 2βπ2π
2(Ξ (π β 12
)
Ξ (π2))
2
ππ π > 2
Other comments π has a standard non-central π‘ distribution with π degrees of
freedom and non-centrality parameter π if π = (π + π) βπ πββ , π~ππ
2, π~π(0,1) and π and π are independent. It has the following non-central moments:
πΈ(ππ) = (π
2)
π2 Ξ (
π β π2
)
Ξ (π2)
πβπ2 2β
ππππ2 2β
ππ ππ π > π
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βnon-central tβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βnon-central t4β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The normal distribution [NormalDistribution] The normal distribution is a continuous probability distribution that has a bell-shaped probability density function:
π(π₯) =1
πβ2πexp (β
1
2(π₯ β π
π)2
)
It is usually considered to be the most prominent probability distribution in statistics partly because it arises in a very large number of contexts as a result of the central limit theorem and partly because it is relatively tractable analytically. The normal distribution is also called the Gaussian distribution. The unit normal (or standard normal) distribution is π(0,1). Characteristics of the normal distribution are set out below:
Distribution name Normal distribution
Common notation π~π(π, π2) Parameters π = scale parameter (π > 0)
π = location parameter
Domain ββ < π₯ < +β
Probability density function π(π₯) β‘ π(π₯) =
1
πβ2πexp (β
1
2(π₯ β π
π)2
)
Cumulative distribution function πΉ(π₯) β‘ π(π₯) = Ξ¦(π₯) =
1
β2πβ«exp (β
1
2(π‘ β π
π)2
)ππ‘
π₯
ββ
Mean π Variance π2 Skewness 0
(Excess) kurtosis 0
Characteristic function πππ‘πβ
12π2π‘2
Other comments The inverse unit normal distribution function (i.e. its quantile function) is commonly written πβ1(π₯) (also in some texts Ξ¦(π₯) and the unit normal density function is commonly written π(π₯). πβ1(π₯) is also called the probit function.
The error function distribution is π (0,1
2β), where β is now an inverse
scale parameter β > 0. The median and mode of a normal distribution are π. The truncated first moments of π(π, π2) are:
β« π₯π(π₯)ππ₯
π
πΏ
= π (π (π β π
π) β π (
πΏ β π
π))
β π (π (π β π
π) β π (
πΏ β π
π))
where π(π₯) and π(π₯) are the pdf and cdf of the unit normal distribution respectively. The mean excess function of a standard normal distribution is thus
π(π’) =π(π’) β π’π(βπ’)
π(βπ’)=
1
β2πππ₯π (β
12π’2) β π’π(βπ’)
π(βπ’)
The central moments of the normal distribution are:
πΈ((π β π)π) = {0 ππ π ππ πππ
ππΓ1Γ3Γβ¦Γ(π β 1) ππ π ππ ππ£ππ
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βnormalβ. For details of other supported probability distributions see here.
The Pareto distribution [ParetoDistribution]
Distribution name Pareto distribution
Common notation π~πππππ‘π(πΌ, π½) Parameters πΌ = shape parameter (πΌ > 0)
π½ = scale parameter (π½ > 0)
Domain 0 β€ π₯ < +β Probability density function
π(π₯) =πΌπ½πΌ
(π½ + π₯)πΌ+1
Cumulative distribution function πΉ(π₯) = 1 β (
π½
π½ + π₯)πΌ
Mean π½
πΌ β 1 πππ πΌ > 1
Variance πΌπ½2
(πΌ β 1)2(πΌ β 2) πππ πΌ > 2
Skewness 2(πΌ + 1)
πΌ β 3βπΌ β 2
πΌ πππ πΌ > 3
(Excess) kurtosis 6(πΌ3 + πΌ2 β 6πΌ β 2)
πΌ(πΌ β 3)(πΌ β 4) πππ πΌ > 4
Other comments Also known as the Pareto distribution of the second kind in which case the Pareto distribution of the first kind has π½ β€ π₯ < +β,
π(π₯) =πΌπ½πΌ
π₯πΌ+1 and πΉ(π₯) = 1 β (
π½
π₯)πΌ
.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βparetoβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βpareto3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The power function distribution [PowerFunctionDistribution]
Distribution name Power function distribution
Common notation
Parameters π = shape parameter (π > 0)
Domain 0 β€ π₯ β€ 1
Probability density function π(π₯) = ππ₯πβ1 Cumulative distribution function
πΉ(π₯) = π₯π
Mean π
π + 1
Variance π
π3 + 4π2 + 5π + 2
Other comments Its non-central moments around π (which can be used to derive its non-central moments around 0) are:
πΈ(ππ) =π
π + π
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βpowerβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of βpower3β β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The Rayleigh distribution [RayleighDistribution]
Distribution name Rayleigh distribution
Common notation π~π ππ¦ππππβ(π)
Parameters π = scale parameter (π > 0)
Domain 0 β€ π₯ < +β
Probability density function
π(π₯) =π₯ exp (β
π₯2
2π2)
π2
Cumulative distribution function πΉ(π₯) = 1 β exp(β
π₯2
2π2)
Mean πβπ
2
Variance 4 β π
2π2
Skewness 2βπ(π β 3)
(4 β π)2
(Excess) kurtosis β6π2 β 24π + 16
(4 β π)2
Characteristic function 1 β ππ‘ exp (β
π2π‘2
2)β
π
2(βπ erf (
πππ‘
β2) β π)
Here erf(π§) is the error function, which is defined as:
erf(π§) =2
βπβ«πβπ‘
2ππ‘
π§
0
βΉ erf(π₯) = 2.π(π₯β2) β 1
Other comments The Rayleigh distribution often arises when the overall magnitude of a vector is related to its directional components
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βrayleighβ. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of βrayleigh2β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The reciprocal distribution [ReciprocalDistribution]
Distribution name Reciprocal distribution
Common notation
Parameters π, π = boundary parameters (0 < π < π)
Domain π β€ π₯ β€ π
Probability density function π(π₯) =
1
π₯(log π β log π)
Cumulative distribution function
πΉ(π₯) =log π₯ β log π
log π β log π
Mean π β π
log π β log π
Variance 1
2(
π2 β π2
log π β log π) β (
π β π
log π β log π)2
Other comments Its non-central moments are:
πΈ(ππ) =ππ β ππ
π(log π β log π)
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βreciprocalβ. For details of other supported probability distributions see here.
The Rice distribution [RiceDistribution]
The Rice distribution is characterises the magnitude of a circular bivariate normal random vector with potentially non-zero mean.
Distribution name Rice distribution
Common notation π~π πππ(π, π) Parameters π = parameter
π = parameter
Domain 0 β€ π₯ < +β
Probability density function π(π₯) =
π₯
π2exp(β
π₯2 + π2
2π2) πΌ0 (
π₯π
π2)
where πΌ0(π) is the modified Bessel function of the first kind of order
zero, i.e. πΌ0(π) = β(π
2)2π
(π!)2βπ=0
Cumulative distribution function
πΉ(π₯) = 1 β π1 (π
π,π₯
π)
where π1(π, π) is the Marcum Q-function, i.e. π1(π, π) =
β« π₯ exp (βπ₯2+π2
2) πΌ0(ππ₯)ππ₯
β
π.
Mean πβπ 2β πΏ1 2β (β
π2
2π2)
Variance 2π2 + π2 β
ππ2
2(πΏ1 2β (β
π2
2π2))
2
Other comments The Rayleigh distribution is a special case of the Rice distribution where π = 0. Its non-central moments are:
πΈ(ππ) = ππ2π 2β Ξ (1 +π
2) πΏπ 2β (β
π2
2π2) π = 1,2,β¦
where πΏπ(π₯) = πΉ1(βπ; 1; π₯)1 is a Laguerre polynomial.
Nematrian web functions This distribution is not currently supported within the Nematrian web function library. For details of supported probability distributions see here.
The Studentβs t distribution [StudentsTDistribution] The Studentβs t distribution (more simply the t distribution) arises when estimating the mean of a normally distributed population when sample sizes are small and the population standard deviation is unknown.
Distribution name (Standard) Studentβs t distribution
Common notation π~π‘π
Parameters π = degrees of freedom (π > 0, usually π is an integer although in some situations a non-integral π can arise)
Domain ββ < π₯ < +β Probability density function
π(π₯) =1
βππ
Ξ (π + 12 )
Ξ (π2)
(1 +π₯2
π£)
βπ+12
=1
βππ΅ (12,π2)(1 +
π₯2
π£)
βπ+12
Cumulative distribution function
πΉ(π₯) = {
1
2πΌπ§ (
π
2,1
2) π₯ < 0
1 β1
2πΌπ§ (
π
2,1
2) π₯ β₯ 0
where π§ = π (π + π₯2)β
Mean 0
Variance π
π β 2 πππ π > 2
Skewness 0 πππ π > 3
(Excess) kurtosis 3(π β 2)
π β 4β 3 =
6
π β 4 πππ π > 4
Characteristic function πΎπ 2β (βπ|π‘|)(βπ|π‘|)π 2β
Ξ(π 2β )2π 2β β1
where πΎπ 2β (π₯) is a Bessel function
Other comments It is a special case of the generalised hyperbolic distribution. Its non-central moments if π is even and 0 < π < π are:
πΈ(ππ) =Ξ (π + 12 ) Ξ (
π β π2 ) ππ 2β
βπΞ(π2)
If π is even and 0 < π β€ π then πΈ(ππ) = β, if π is odd and 0 < π < π then πΈ(ππ) = 0 and if π is odd and 0 < π β€ π then πΈ(ππ) is undefined.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βstudent's tβ. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a
DistributionName of βstudent's t3β, see also including additional shift and scale parameters. For details of other supported probability distributions see here.
The triangular distribution [TriangularDistribution]
Distribution name Triangular distribution
Common notation π~πππππππ’πππ(π, π, π)
Parameters π, π = boundary parameters (π < π) π = mode parameter (π β€ π β€ π)
Domain π β€ π₯ β€ π
Probability density function
π(π₯) =
{
2(π₯ β π)
(π β π)(π β π)π β€ π₯ β€ π
2(π β π₯)
(π β π)(π β π)π < π₯ β€ π
Cumulative distribution function
πΉ(π₯) =
{
(π₯ β π)2
(π β π)(π β π)π β€ π₯ β€ π
1 β(π β π₯)2
(π β π)(π β π)π < π₯ β€ π
Mean π + π + π
3
Variance π2 + π2 + π2 β ππ β ππ β ππ
18
Skewness β2(π + π β 2π)(2π β π β π)(π β 2π + π)
5(π2 + π2 + π2 β ππ β ππ β ππ)3 2β
(Excess) kurtosis β3
5
Characteristic function β2
(π β π)ππππ‘ β (π β π)ππππ‘ + (π β π)ππππ‘
(π β π)(π β π)(π β π)π‘2
Other comments Its mode is π. Its median is:
{
π + β
(π β π)(π β π)
2π β₯
π + π
2
π β β(π β π)(π β π)
2π β€
π + π
2
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βtriangularβ. For details of other supported probability distributions see here.
The uniform distribution [UniformDistribution] The uniform distribution describes a (continuous) probability distribution in which any outcome within a given range is equally probable.
Distribution name Uniform distribution
Common notation π~π(π, π) Parameters π, π = boundary parameters (π < π)
Domain π β€ π₯ β€ π
Probability density function π(π₯) =
1
π β π
Cumulative distribution function
πΉ(π₯) =π₯ β π
π β π
Mean (π + π) 2β Variance (π β π)2 12β Skewness 0
(Excess) kurtosis β6 5β
Characteristic function ππππ‘ β ππππ‘
ππ‘(π β π)
Other comments Its non-central moments (π = 1,2,3, β¦ ) are πΈ(ππ) =1
(πβ1)
1
π+1(ππ+1 β ππ+1). Its median is (π + π) 2β .
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βuniformβ. For details of other supported probability distributions see here.
The Weibull distribution
[WeibullDistribution] The Weibull distribution is a special case of the generalised extreme value (GEV) distribution. It characterises the distribution of βblock maximaβ under certain (relatively restrictive) conditions.
Continuous multivariate distributions
The inverse Wishart distribution [InverseWishartDistribution] The inverse Wishart distribution (otherwise called the inverted Wishart distribution) πβ1(π,π) is a probability distribution that is used in the Bayesian analysis of real-valued positive definite matrices (e.g. matrices of the type that arise in risk management contexts). It is a conjugate prior for the covariance matrix of a multivariate normal distribution. It has the following characteristics, where π is a πΓπ matrix, π is a positive definite matrix and Ξπ(. ) is the multivariate gamma function.
Parameters (and constraints on parameters): π > π β 1 (π = degrees of freedom, real) π > 0 (π = inverse scale matrix, positive definite)
Support (i.e. values that it can take) π > 0, i.e. is positive definite, π an πΓπ matrix
Probability density function |π|π 2β |π|β(π+π+1) 2β πβtrace(ππβ1) 2β
2ππ 2β Ξπ(π 2β )
Mean π
π β π β 1
If the elements of π are π΅π,π and the elements of π are ππ,π then
var(π΅π,π) =(π β π + 1)ππ,π
2 + (π β π β 1)ππ,πππ,π
(π β π)(π β π β 1)2(π β π β 3)
The main use of the inverse Wishart distribution appears to arise in Bayesian statistics. Suppose we want to make an inference about a covariance matrix, π, whose prior π(π) has a πβ1(π,π) distribution. If the observation set π = (π±1, β¦ , π±πΎ) where the ππ are independent π-variate Normal (i.e. Gaussian) random variables drawn from a π(0, π) distribution then the conditional distribution π(π|π), i.e. the probability of π given π, has a πβ1(π + π,π + πΎ) distribution, where π = πππ is the sample covariance matrix. The univariate special case of the inverse Wishart distribution is the inverse gamma distribution. With π = 1, πΌ = π 2β , π½(= π 2β ) = π1,1 2β , π₯(= π 2β ) = π΅1,1 2β we have:
π(π₯|πΌ, π½) =π½πΌπ₯βπΌβ1exp(βΞ² xβ )
Ξ(Ξ±)
where Ξ(. ) β‘ Ξ1(. ) is the ordinary (i.e. univariate) Gamma function, see MnGamma. For other probability distributions see here.
Copulas (a copula is a special type of continuous multivariate distribution)
The Clayton copula [ClaytonCopula] The Clayton copula is a copula that allows any specific non-zero level of (lower) tail dependency between individual variables. It is an Archimedean copula, and exchangeable.
Copula name Clayton copula
Common notation π~πΆππΆπ
Parameters π β₯ 0 (can be extended to π β₯ β1
Domain 0 β€ π’π β€ 1 π = 1,β¦ , π
Copula
πΆππΆπ(π’1, β¦ , π’π) = (β(π’1
βπ β 1)
π
+ 1)
β1 πβ
Or if π = 0 we use the limit limπβ0
πΆππΊπΆ
Kendallβs rank correlation coefficient (for bivariate case)
π
2 + π
Coefficient of upper tail dependence, ππ’
0
Coefficient of lower tail dependence, ππ
2β1 (ππΏ)β
Archimedean generator function, π(π‘)
πβπΏ(π‘βπ β 1)
Or if π = 0 we use the limit limπβ0
ππ(π‘) = β log π‘
Other comments If π = 0 we obtain the independence copula. The Clayton copula (like the Frank copula) is a comprehensive copula in that it interpolates between a lower limit of the countermonotonicity copula (π β β1) and an upper limit of the comonotonicity copula (π β +β)..
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βClayton Copulaβ. For details of other probability distributions see here.
The comonotonicity copula [ComonotonicityCopula] The Comonotonicity copula is a special copula characterising perfect positive dependence, in the sense that the ππ are almost surely strictly increasing functions of each other.
Copula name comonotonicity copula
Common notation π~π Parameters π
Domain 0 β€ π’π β€ 1 π = 1,β¦ , π Copula π(π’1, β¦ , π’π) = πππ(π’1, β¦ π’π)
Kendallβs rank correlation coefficient (for bivariate case)
1
Coefficient of upper tail dependence
1
Coefficient of lower tail dependence
1
Other comments Is the extreme case where dependency is as strongly βpositiveβ as possible (i.e. achieves the FrΓ©chet upper bound)
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βComonotonicity Copulaβ. For details of other supported probability distributions see here.
The countermonotonicity copula [CountermonotonicityCopula] The Countermonotonicity copula is a special copula characterising perfect negative dependence, in the sense that π1 is almost surely a strictly decreasing function of π2 and vice-versa.
Copula name countermonotonicity copula
Common notation π~π
Parameters π Domain 0 β€ π’π β€ 1 π = 1,2
Copula π(π’1, π’2) = max(π’1 + π’2 β 1,0) Kendallβs rank correlation coefficient
β1
Coefficient of upper tail dependence
1
Coefficient of lower tail dependence
1
Other comments Is the extreme case where dependency is as strongly βnegativeβ as possible (i.e. achieves the FrΓ©chet lower bound)
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βCountermonotonicity Copulaβ. For details of other supported probability distributions see here.
The Frank copula [FrankCopula] The Frank copula is a copula that is sometimes used in the modelling of codependency. It is an Archimedean copula, and exchangeable.
Copula name Frank copula
Common notation π~πΆππΉπ
Parameters ββ < π < β Domain 0 β€ π’π β€ 1 π = 1,β¦ , π Copula
πΆππΉπ(π’1, β¦ , π’π) = β
1
πlog (1 +
β (exp(βππ’π) β 1)π
exp(βπ) β 1)
Or if π = 0 we use the limit limπβ0
πΆππΉπ which the independence copula
Kendallβs rank correlation coefficient (for bivariate case)
1 β 4πβ1(1 β π·1(π))
where π·1(π) is the Debye function defined as:
π·1(π) = πβ1β«
π‘
exp(π‘) β 1ππ‘
β
0
Coefficient of upper tail dependence, ππ’
0
Coefficient of lower tail dependence, ππ
0
Archimedean generator function, π(π‘) β log(
exp(βππ‘) β 1
exp(βπ) β 1)
Or if π = 0 we use the limit limπβ0
ππ(π‘) which is taken as β log π‘
Other comments If π = 0 we obtain the independence copula. The Frank copula (like the Clayton copula) is a comprehensive copula in that it interpolates between a lower limit of the countermonotonicity copula (π β ββ) and an upper limit of the comonotonicity copula (π β +β).
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βFrank Copulaβ. For details of other supported probability distributions see here.
The Generalised Clayton copula [GeneralisedClaytonCopula] The Generalised Clayton copula is a copula that allows any specific (non-zero) level of (lower) tail dependency between individual variables. It is an Archimedean copula and exchangeable.
Copula name Generalised Clayton copula
Common notation π~πΆπ,πΏπΊπΆ
Parameters π β₯ 0, πΏ β₯ 1
Domain 0 β€ π’π β€ 1 π = 1,β¦ , π Copula
πΆπ,πΏπΊπΆ(π’1, β¦ , π’π) = ((β(π’1
βπ β 1)πΏ
π
)
1 πΏβ
+ 1)
β1 πβ
Or if π = 0 we use the limit limπβ0
πΆππΊπΆ
Kendallβs rank correlation coefficient (for bivariate case), ππ
(2 + π)πΏ β 2
(2 + π)πΏ
Coefficient of upper tail 2 β 2β1 πΏβ
dependence, ππ’
Coefficient of lower tail dependence, ππ
2β1 (ππΏ)β
Archimedean generator function, π(π‘)
πβπΏ(π‘βπ β 1)
Or if π = 0 we use the limit limπβ0
ππ(π‘)
Other comments When πΏ = 1 the generalised Clayton copula becomes the Clayton copula.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βGeneralised Clayton Copulaβ. For details of other supported probability distributions see here.
The Gumbel copula [GumbelCopula] The Gumbel copula is a copula that allows any specific level of (upper) tail dependency between individual variables. It is an Archimedean copula, and exchangeable.
Copula name Gumbel copula
Common notation π~πΆππΊπ’
Parameters 1 β€ π < β
Domain 0 β€ π’π β€ 1 π = 1,β¦ , π Copula
πΆππΊπ’(π’1, β¦ , π’π) = exp(β(β(β logπ’π)
π
π
)
1 πβ
)
Kendallβs rank correlation coefficient (for bivariate case)
1 β1
π
Coefficient of upper tail dependence, ππ’
2 β 21 πβ
Coefficient of lower tail dependence, ππ
0
Archimedean generator function, π(π‘)
(β log π‘)π
Other comments If π = 1 we obtain the independence copula and as π β β we approach the comonotonicity copula.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βGumbel Copulaβ. For details of other supported probability distributions see here.
The Gaussian copula [GaussianCopula]
The Gaussian copula is the copula that underlies the multivariate normal distribution.
Copula name Gaussian copula
Common notation π~πΆπΆπΊπ
Parameters πΆ, a non-negative definite πΓπ matrix, i.e. a matrix that can correspond to a correlation matrix
Domain 0 β€ π’π β€ 1 π = 1,β¦ , π Copula πΆπΆ
πΊπ(π’1, β¦ , π’π) = π½πΆ(πβ1(π’1),β¦ ,π
β1(π’2))
where πβ1(π₯) is the inverse normal function and π½πΆ is the cumulative distribution function of the multivariate normal distribution defined by a covariance matrix equal to πΆ
Kendallβs rank correlation coefficient (for bivariate case), ππ
2
πarcsinπ
Where π is the correlation coefficient between the two variables
Coefficient of upper tail dependence, ππ’
0 (unless the correlation matrix exhibits perfect positive or negative dependence)
Coefficient of lower tail dependence, ππ
0 (unless the correlation matrix exhibits perfect positive or negative dependence)
Other comments The Spearman rank correlation coefficient is given by:
ππ =6
πarcsin
π
2
where π is the (normal) correlation coefficient between the two variables. If πΆ = πΌπ (the πΓπ identity matrix) then we obtain the independence copula.
Nematrian web functions Functions relating to the above distribution in the two dimensional case may be accessed via the Nematrian web function library by using a DistributionName of βGaussian Copula (2d)β. For details of other supported probability distributions see here.
The Independence copula [IndependenceCopula] The Independence copula is the copula that results from a dependency structure in which each individual variable is independent of each other. It is an Archimedean copula, and exchangeable.
Copula name Independence copula
Common notation π~Ξ
Parameters None
Domain 0 β€ π’π β€ 1 π = 1,β¦ , π Copula Ξ (π’1, β¦ , π’π) =βπ’π
π
Kendallβs rank correlation coefficient (for bivariate case)
0
Coefficient of upper tail 0
dependence, ππ’
Coefficient of lower tail dependence, ππ
0
Archimedean generator function, π(π‘)
β log π‘
Other comments The independence copula is a special case of several Archimedean copulas. It is also the special case of the Gaussian copula with a correlation matrix equal to the identity matrix.
Nematrian web functions Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of βIndependence Copulaβ. For details of other supported probability distributions see here.
The t copula [TCopula] The t copula is the copula that underlies the multivariate Studentβs t distribution.
Copula name t copula
Common notation π~πΆπ,πΆπ‘
Parameters πΆ, a non-negative definite πΓπ matrix, i.e. a matrix that can correspond to a correlation matrix
π = degrees of freedom (π > 0, usually π is an integer although in some situations a non-integral π can arise)
(note in principle each marginal distribution could in principle have a different number of degrees of freedom although such a refinement
is not commonly seen)
Domain 0 β€ π’π β€ 1 π = 1,β¦ , π
Copula πΆπ,πΆπ‘ (π’1, β¦ , π’π) = ππ,πΆ(π‘π
β1(π’1), β¦ , π‘πβ1(π’2))
where π‘πβ1(π₯) is the inverse studentβs t function and ππ,πΆ is the
cumulative distribution function of the multivariate studentβs t distribution with arbitrary mean and matrix generator equal to πΆ
Kendallβs rank correlation coefficient (for bivariate case), ππ
2
πarcsinπ
Where π is the correlation coefficient between the two variables
Coefficient of upper tail dependence, ππ’ 2π‘π+1(ββ
(π + 1)(1 β π)
1 + π)
Coefficient of lower tail dependence, ππ 2π‘π+1(ββ
(π + 1)(1 β π)
1 + π)
Other comments If πΆ = πΌπ (the πΓπ identity matrix) then, in contrast to the Gaussian copula, we do not recover the independence copula.
Nematrian web functions
Functions relating to the above distribution in the two-dimensional case may be accessed via the Nematrian web function library by using a DistributionName of βstudentβs t (2d)β. For details of other supported probability distributions see here.