# Continuous Statistical Distributions¶

## Overview¶

All distributions will have location (L) and Scale (S) parameters along with any shape parameters needed, the names for the shape parameters will vary. Standard form for the distributions will be given where and The nonstandard forms can be obtained for the various functions using (note is a standard uniform random variate).

Function Name Standard Function Transformation
Cumulative Distribution Function (CDF)
Probability Density Function (PDF)
Percent Point Function (PPF)
Probability Sparsity Function (PSF)
Hazard Function (HF)
Cumulative Hazard Functon (CHF)
Survival Function (SF)
Inverse Survival Function (ISF)
Moment Generating Function (MGF)
Random Variates
(Differential) Entropy
(Non-central) Moments
Central Moments
mean (mode, median), var
skewness, kurtosis

### Moments¶

Non-central moments are defined using the PDF

Note, that these can always be computed using the PPF. Substitute in the above equation and get

which may be easier to compute numerically. Note that so that Central moments are computed similarly

In particular

Skewness is defined as

while (Fisher) kurtosis is

so that a normal distribution has a kurtosis of zero.

### Median and mode¶

The median, is defined as the point at which half of the density is on one side and half on the other. In other words, so that

In addition, the mode, , is defined as the value for which the probability density function reaches it’s peak

### Fitting data¶

To fit data to a distribution, maximizing the likelihood function is common. Alternatively, some distributions have well-known minimum variance unbiased estimators. These will be chosen by default, but the likelihood function will always be available for minimizing.

If is the PDF of a random-variable where is a vector of parameters ( e.g. and ), then for a collection of independent samples from this distribution, the joint distribution the random vector is

The maximum likelihood estimate of the parameters are the parameters which maximize this function with fixed and given by the data:

Where

Note that if includes only shape parameters, the location and scale-parameters can be fit by replacing with in the log-likelihood function adding and minimizing, thus

If desired, sample estimates for and (not necessarily maximum likelihood estimates) can be obtained from samples estimates of the mean and variance using

where and are assumed known as the mean and variance of the untransformed distribution (when and ) and

### Standard notation for mean¶

We will use

where should be clear from context as the number of samples

## Alpha¶

One shape parameters (parameter in DATAPLOT is a scale-parameter). Standard form is

No moments?

Defined over

## Arcsine¶

Defined over . To get the JKB definition put i.e. and

## Beta¶

Two shape parameters

is also called the Power-function distribution.

All of the

## Beta Prime¶

Defined over (Note the CDF evaluation uses Eq. 3.194.1 on pg. 313 of Gradshteyn & Ryzhik (sixth edition).

Therefore,

where is the exponential integral function. Also

## Cauchy¶

No finite moments. This is the t distribution with one degree of freedom.

## Chi¶

Generated by taking the (positive) square-root of chi-squared variates.

## Chi-squared¶

This is the gamma distribution with and and where is called the degrees of freedom. If are all standard normal distributions, then has (standard) chi-square distribution with degrees of freedom.

The standard form (most often used in standard form only) is

## Cosine¶

Approximation to the normal distribution.

## Double Gamma¶

The double gamma is the signed version of the Gamma distribution. For

## Double Weibull¶

This is a signed form of the Weibull distribution.

## Erlang¶

This is just the Gamma distribution with shape parameter an integer.

## Exponential¶

This is a special case of the Gamma (and Erlang) distributions with shape parameter and the same location and scale parameters. The standard form is therefore ( )

## Exponentiated Weibull¶

Two positive shape parameters and and

## Exponential Power¶

One positive shape parameter . Defined for

## Fatigue Life (Birnbaum-Sanders)¶

This distribution’s pdf is the average of the inverse-Gaussian and reciprocal inverse-Gaussian pdf . We follow the notation of JKB here with for

## Fisk (Log Logistic)¶

Special case of the Burr distribution with

## Folded Cauchy¶

This formula can be expressed in terms of the standard formulas for the Cauchy distribution (call the cdf and the pdf ). if is cauchy then is folded cauchy. Note that

No moments

## Folded Normal¶

If is Normal with mean and , then is a folded normal with shape parameter , location parameter and scale parameter . This is a special case of the non-central chi distribution with one- degree of freedom and non-centrality parameter Note that . The standard form of the folded normal is

## Fratio (or F)¶

Defined for . The distribution of if is chi-squared with degrees of freedom and is chi-squared with degrees of freedom.

## Fréchet (ExtremeLB, Extreme Value II, Weibull minimum)¶

A type of extreme-value distribution with a lower bound. Defined for and

where is Euler’s constant and equal to

## Fréchet (left-skewed, Extreme Value Type III, Weibull maximum)¶

Defined for and .

The mean is the negative of the right-skewed Frechet distribution given above, and the other statistical parameters can be computed from

where is Euler’s constant and equal to

## Gamma¶

The standard form for the gamma distribution is valid for .

where

## Generalized Logistic¶

Has been used in the analysis of extreme values. Has one shape parameter And

Note that the polygamma function is

where is a generalization of the Riemann zeta function called the Hurwitz zeta function Note that

## Generalized Pareto¶

Shape parameter and defined for for all and if is negative.

Thus,

## Generalized Exponential¶

Three positive shape parameters for Note that and are all

## Generalized Extreme Value¶

Extreme value distributions with shape parameter .

For defined on

So,

For defined on For defined over all space

This is just the (left-skewed) Gumbel distribution for c=0.

## Generalized Gamma¶

A general probability form that reduces to many common distributions: and

Special cases are Weibull , half-normal and ordinary gamma distributions If then it is the inverted gamma distribution.

For and we have

## Gilbrat¶

Special case of the log-normal with and (typically also )

## Gompertz (Truncated Gumbel)¶

For and . In JKB the two shape parameters are reduced to the single shape-parameter . As is just a scale parameter when . If the distribution reduces to the exponential distribution scaled by Thus, the standard form is given as

where

## Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value)¶

One of a clase of extreme value distributions (right-skewed).

## Gumbel Left-skewed (for minimum order statistic)¶

Note, that is negative the mean for the right-skewed distribution. Similar for median and mode. All other moments are the same.

## HalfCauchy¶

If is Hyperbolic Secant distributed then is Half-Cauchy distributed. Also, if is (standard) Cauchy distributed, then is Half-Cauchy distributed. Special case of the Folded Cauchy distribution with The standard form is

No moments, as the integrals diverge.

## HalfNormal¶

This is a special case of the chi distribution with and and This is also a special case of the folded normal with shape parameter and If is (standard) normally distributed then, is half-normal. The standard form is

## Half-Logistic¶

In the limit as for the generalized half-logistic we have the half-logistic defined over Also, the distribution of where has logistic distribtution.

## Hyperbolic Secant¶

Related to the logistic distribution and used in lifetime analysis. Standard form is (defined over all )

where is an integer given by

where is the Bernoulli polynomial of order evaluated at Thus

,

## Inverted Gamma¶

Special case of the generalized Gamma distribution with and ,

## Inverse Normal (Inverse Gaussian)¶

The standard form involves the shape parameter (in most definitions, is used). (In terms of the regress documentation ) and and is not a parameter in that distribution. A standard form is

This is related to the canonical form or JKB “two-parameter “inverse Gaussian when written in it’s full form with scale parameter and location parameter by taking and then is equal to where is the parameter used by JKB. We prefer this form because of it’s consistent use of the scale parameter. Notice that in JKB the skew and the kurtosis ( ) are both functions only of as shown here, while the variance and mean of the standard form here are transformed appropriately.

## Inverted Weibull¶

Shape parameter and . Then

where is Euler’s constant.

## Johnson SB¶

Defined for with two shape parameters and

## Johnson SU¶

Defined for all with two shape parameters and .

## Laplace (Double Exponential, Bilateral Expoooonential)¶

The ML estimator of the location parameter is

where is a sequence of mutually independent Laplace RV’s and the median is some number between the and the order statistic ( e.g. take the average of these two) when is even. Also,

Replace with if it is known. If is known then this estimator is distributed as .

## Left-skewed Lévy¶

Special case of Lévy-stable distribution with and the support is . In standard form

No moments.

## Lévy¶

A special case of Lévy-stable distributions with and . In standard form it is defined for as

It has no finite moments.

## Logistic (Sech-squared)¶

A special case of the Generalized Logistic distribution with Defined for

## Log Double Exponential (Log-Laplace)¶

Defined over with

## Log Gamma¶

A single shape parameter (Defined for all )

## Log Normal (Cobb-Douglass)¶

Has one shape parameter >0. (Notice that the “Regress “ where is the scale parameter and is the mean of the underlying normal distribution). The standard form is

Notice that using JKB notation we have and we have given the so-called antilognormal form of the distribution. This is more consistent with the location, scale parameter description of general probability distributions.

Also, note that if is a log-normally distributed random-variable with and and shape parameter Then, is normally distributed with variance and mean

## Nakagami¶

Generalization of the chi distribution. Shape parameter is Defined for

## Noncentral beta*¶

Defined over with and and

## Noncentral chi-squared¶

The distribution of where are independent standard normal variables and are constants. (In communications it is called the Marcum-Q function). Can be thought of as a Generalized Rayleigh-Rice distribution. For

Let and and

## Noncentral t¶

The distribution of the ratio

where and are independent and distributed as a standard normal and chi with degrees of freedom. Note and .

## Maxwell¶

This is a special case of the Chi distribution with and and

## Mielke’s Beta-Kappa¶

A generalized F distribution. Two shape parameters and , and . The in the DATAPLOT reference is a scale parameter.

## Pareto¶

For and . Standard form is

## Pareto Second Kind (Lomax)¶

This is Pareto of the first kind with so

## Power Log Normal¶

A generalization of the log-normal distribution and and

This distribution reduces to the log-normal distribution when

## Power Normal¶

A generalization of the normal distribution, for

For this reduces to the normal distribution.

## Power-function¶

A special case of the beta distribution with : defined for

## R-distribution¶

A general-purpose distribution with a variety of shapes controlled by Range of standard distribution is

The R-distribution with parameter is the distribution of the correlation coefficient of a random sample of size drawn from a bivariate normal distribution with The mean of the standard distribution is always zero and as the sample size grows, the distribution’s mass concentrates more closely about this mean.

## Rayleigh¶

This is Chi distribution with and and (no location parameter is generally used), the mode of the distribution is

Defined for and

Shape parameters

## Reciprocal Inverse Gaussian¶

The pdf is found from the inverse gaussian (IG), defined for as

Defined on

## Student t¶

Shape parameter is the incomplete beta integral and

As this distribution approaches the standard normal distribution.

where

## Student Z¶

The student Z distriubtion is defined over all space with one shape parameter

Interesting moments are

The moment generating function is

## Triangular¶

One shape parameter giving the distance to the peak as a percentage of the total extent of the non-zero portion. The location parameter is the start of the non- zero portion, and the scale-parameter is the width of the non-zero portion. In standard form we have

## Truncated Exponential¶

This is an exponential distribution defined only over a certain region . In standard form this is

## Truncated Normal¶

A normal distribution restricted to lie within a certain range given by two parameters and . Notice that this and correspond to the bounds on in standard form. For we get

where

Notice that the

## Uniform¶

Standard form In general form, the lower limit is the upper limit is

## Von Mises¶

Defined for with shape parameter . Note, the PDF and CDF functions are periodic and are always defined over regardless of the location parameter. Thus, if an input beyond this range is given, it is converted to the equivalent angle in this range. For values of the PDF and CDF formulas below are used. Otherwise, a normal approximation with variance is used.

This can be used for defining circular variance.

## Wald¶

Special case of the Inverse Normal with shape parameter set to . Defined for .

For