Statistics: InverseGaussianDistribution

Class Definition: `InverseGaussianDistribution`

statistics: InverseGaussianDistribution

Inverse Gaussian probability distribution object.

A InverseGaussianDistribution object consists of parameters, a model description, and sample data for a Inverse Gaussian probability distribution.

The Inverse Gaussian distribution is a continuous probability distribution, which is often used to model non-negative positively skewed data. Is is defined by mean parameter mu and shape parameter lambda.

There are several ways to create a InverseGaussianDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor InverseGaussianDistribution (mu, lambda) to create a Inverse Gaussian distribution with fixed parameter values mu and lambda.
Use the static method InverseGaussianDistribution.fit (x, alpha, censor, freq, options) to fit a distribution to the data in x using the same input arguments as the invgfit function.

It is highly recommended to use fitdist and makedist functions to create probability distribution objects, instead of the class constructor or the aforementioned static method.

Further information about the Inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: fitdist, makedist, invgcdf, invginv, invgpdf, invgrnd, invgfit, invglike, invgstat

Source Code: InverseGaussianDistribution

The InverseGaussianDistribution class contains the following properties:

A positive scalar value characterizing the mean of the Inverse Gaussian distribution. You can access the mu property using dot name assignment.

Example: 1

Create an Inverse Gaussian distribution with default parameters

 pd = makedist ("InverseGaussian")

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 1

Query parameter 'mu' (mean parameter)

 pd.mu

ans = 1

Set parameter 'mu'

 pd.mu = 2

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 2
   lambda = 1

Use this to initialize or modify the mean parameter of an Inverse Gaussian distribution. The mean parameter must be a positive real scalar.

Example: 2

Create an Inverse Gaussian distribution object by calling its constructor

 pd = InverseGaussianDistribution (1.5, 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1.5
   lambda =   2

Query parameter 'mu'

 pd.mu

ans = 1.5000

This demonstrates direct construction with a specific mean parameter, useful for modeling non-negative skewed data with a known mean.

A positive scalar value characterizing the shape of the Inverse Gaussian distribution. You can access the lambda property using dot name assignment.

Example: 1

Create an Inverse Gaussian distribution with default parameters

 pd = makedist ("InverseGaussian")

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 1

Query parameter 'lambda' (shape parameter)

 pd.lambda

ans = 1

Set parameter 'lambda'

 pd.lambda = 3

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 3

Use this to initialize or modify the shape parameter in an Inverse Gaussian distribution. The shape parameter must be a positive real scalar.

Example: 2

Create an Inverse Gaussian distribution object by calling its constructor

 pd = InverseGaussianDistribution (1.5, 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1.5
   lambda =   2

Query parameter 'lambda'

 pd.lambda

ans = 2

This shows how to set the shape parameter directly via the constructor, ideal for modeling specific variability in non-negative data.

A character vector specifying the name of the probability distribution object. This property is read-only.

A scalar integer value specifying the number of parameters characterizing the probability distribution. This property is read-only.

A $2×1$ cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A $2×1$ cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A $2×1$ numeric vector containing the values of the distribution parameters. This property is read-only. You can change the distribution parameters by assigning new values to the mu and lambda properties.

A $2×2$ numeric matrix containing the variance-covariance of the parameter estimates. Diagonal elements contain the variance of each estimated parameter, and non-diagonal elements contain the covariance between the parameter estimates. The covariance matrix is only meaningful when the distribution was fitted to data. If the distribution object was created with fixed parameters, or a parameter of a fitted distribution is modified, then all elements of the variance-covariance are zero. This property is read-only.

A $1×2$ logical vector specifying which parameters are fixed and which are estimated. true values correspond to fixed parameters, false values correspond to parameter estimates. This property is read-only.

A $1×2$ numeric vector specifying the truncation interval for the probability distribution. First element contains the lower boundary, second element contains the upper boundary. This property is read-only. You can only truncate a probability distribution with the truncate method.

A logical scalar value specifying whether a probability distribution is truncated or not. This property is read-only.

A scalar structure containing the following fields:

data: a numeric vector containing the data used for distribution fitting.
cens: a numeric vector of logical values indicating censoring information corresponding to the elements of the data used for distribution fitting. If no censoring vector was used for distribution fitting, then this field defaults to an empty array.
freq: a numeric vector of non-negative integer values containing the frequency information corresponding to the elements of the data used for distribution fitting. If no frequency vector was used for distribution fitting, then this field defaults to an empty array.

The InverseGaussianDistribution class offers the following public methods:

InverseGaussianDistribution: p = cdf (pd, x)
InverseGaussianDistribution: p = cdf (pd, x, 'upper')

p = cdf (pd, x) computes the CDF of the probability distribution object, pd, evaluated at the values in x.

p = cdf (…, 'upper') returns the complement of the CDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1

Plot various CDFs from the Inverse Gaussian distribution

 x = 0:0.01:5;
 pd1 = makedist ("InverseGaussian", "mu", 1, "lambda", 1);
 pd2 = makedist ("InverseGaussian", "mu", 1, "lambda", 2);
 pd3 = makedist ("InverseGaussian", "mu", 1, "lambda", 3);
 p1 = cdf (pd1, x);
 p2 = cdf (pd2, x);
 p3 = cdf (pd3, x);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r")
 grid on
 legend ({"mu = 1, lambda = 1", "mu = 1, lambda = 2", "mu = 1, lambda = 3"}, ...
         "location", "southeast")
 title ("Inverse Gaussian CDF")
 xlabel ("values")
 ylabel ("Cumulative probability")

Use this to compute and visualize the cumulative distribution function for different Inverse Gaussian distributions, showing how probability accumulates over values.

InverseGaussianDistribution: x = icdf (pd, p)

x = icdf (pd, p) computes the quantile (the inverse of the CDF) of the probability distribution object, pd, evaluated at the values in p.

Example: 1

Plot various iCDFs from the Inverse Gaussian distribution

 p = 0.001:0.001:0.999;
 pd1 = makedist ("InverseGaussian", "mu", 1, "lambda", 1);
 pd2 = makedist ("InverseGaussian", "mu", 1, "lambda", 2);
 pd3 = makedist ("InverseGaussian", "mu", 1, "lambda", 3);
 x1 = icdf (pd1, p);
 x2 = icdf (pd2, p);
 x3 = icdf (pd3, p);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 legend ({"mu = 1, lambda = 1", "mu = 1, lambda = 2", "mu = 1, lambda = 3"}, ...
         "location", "northwest")
 title ("Inverse Gaussian iCDF")
 xlabel ("Probability")
 ylabel ("values")

This demonstrates the inverse CDF (quantiles) for Inverse Gaussian distributions, useful for finding the value corresponding to given probabilities.

InverseGaussianDistribution: r = iqr (pd)

r = iqr (pd) computes the interquartile range of the probability distribution object, pd.

Example: 1

Compute the interquartile range for an Inverse Gaussian distribution

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 iqr_value = iqr (pd)

iqr_value = 0.7451

Use this to calculate the interquartile range, which measures the spread of the middle 50% of the distribution, useful for understanding variability in non-negative data.

InverseGaussianDistribution: m = mean (pd)

m = mean (pd) computes the mean of the probability distribution object, pd.

Example: 1

Compute the mean for different Inverse Gaussian distributions

 pd1 = makedist ("InverseGaussian", "mu", 1, "lambda", 1);
 pd2 = makedist ("InverseGaussian", "mu", 1, "lambda", 2);
 mean1 = mean (pd1)

mean1 = 1

 mean2 = mean (pd2)

mean2 = 1

This shows how to compute the expected value for Inverse Gaussian distributions with different shape parameters.

InverseGaussianDistribution: m = median (pd)

m = median (pd) computes the median of the probability distribution object, pd.

Example: 1

Compute the median for different Inverse Gaussian distributions

 pd1 = makedist ("InverseGaussian", "mu", 1, "lambda", 1);
 pd2 = makedist ("InverseGaussian", "mu", 1, "lambda", 2);
 median1 = median (pd1)

median1 = 0.6758

 median2 = median (pd2)

median2 = 0.8043

Use this to find the median value, which splits the distribution into two equal probability halves.

InverseGaussianDistribution: nlogL = negloglik (pd)

nlogL = negloglik (pd) computes the negative loglikelihood of the probability distribution object, pd.

Example: 1

Compute the negative loglikelihood for a fitted Inverse Gaussian distribution

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 rand ("seed", 21);
 data = random (pd, 100, 1);
 pd_fitted = fitdist (data, "InverseGaussian")

pd_fitted =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 0.991317   [0.841261, 1.14137]
   lambda =  1.66197   [1.2013, 2.12263]

 nlogL = negloglik (pd_fitted)

nlogL = -77.493

This is useful for assessing the fit of an Inverse Gaussian distribution to data, lower values indicate a better fit.

InverseGaussianDistribution: ci = paramci (pd)
InverseGaussianDistribution: ci = paramci (pd, Name, Value)

ci = paramci (pd) computes the lower and upper boundaries of the 95% confidence interval for each parameter of the probability distribution object, pd.

ci = paramci (pd, Name, Value) computes the confidence intervals with additional options specified by Name-Value pair arguments listed below.

`Name`		`Value`
`'Alpha'`		A scalar value in the range $(0,1)$ specifying the significance level for the confidence interval. The default value 0.05 corresponds to a 95% confidence interval.
`'Parameter'`		A character vector or a cell array of character vectors specifying the parameter names for which to compute confidence intervals. By default, `paramci` computes confidence intervals for all distribution parameters.

paramci is meaningful only when pd is fitted to data, otherwise an empty array, [], is returned.

Example: 1

Compute confidence intervals for parameters of a fitted Inverse Gaussian distribution

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 rand ("seed", 21);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "InverseGaussian")

pd_fitted =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1.00566   [0.961629, 1.0497]
   lambda = 2.01487   [1.83826, 2.19148]

 ci = paramci (pd_fitted, "Alpha", 0.05)

ci =

   0.9616   1.8383
   1.0497   2.1915

Use this to obtain confidence intervals for the estimated parameters (mu and lambda), providing a range of plausible values given the data.

InverseGaussianDistribution: y = pdf (pd, x)

y = pdf (pd, x) computes the PDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1

Plot various PDFs from the Inverse Gaussian distribution

 x = 0:0.01:5;
 pd1 = makedist ("InverseGaussian", "mu", 1, "lambda", 1);
 pd2 = makedist ("InverseGaussian", "mu", 1, "lambda", 2);
 pd3 = makedist ("InverseGaussian", "mu", 1, "lambda", 3);
 y1 = pdf (pd1, x);
 y2 = pdf (pd2, x);
 y3 = pdf (pd3, x);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r")
 grid on
 legend ({"mu = 1, lambda = 1", "mu = 1, lambda = 2", "mu = 1, lambda = 3"}, ...
         "location", "northeast")
 title ("Inverse Gaussian PDF")
 xlabel ("values")
 ylabel ("Probability density")

This visualizes the probability density function for Inverse Gaussian distributions, showing the likelihood of different values.

InverseGaussianDistribution: plot (pd)
InverseGaussianDistribution: plot (pd, Name, Value)
InverseGaussianDistribution: h = plot (…)

plot (pd) plots a probability density function (PDF) of the probability distribution object pd. If pd contains data, which have been fitted by fitdist, the PDF is superimposed over a histogram of the data.

plot (pd, Name, Value) specifies additional options with the Name-Value pair arguments listed below.

`Name`		`Value`
`'PlotType'`		A character vector specifying the plot type. `'pdf'` plots the probability density function (PDF). When `pd` is fit to data, the PDF is superimposed on a histogram of the data. `'cdf'` plots the cumulative density function (CDF). When `pd` is fit to data, the CDF is superimposed over an empirical CDF. `'probability'` plots a probability plot using a CDF of the data and a CDF of the fitted probability distribution. This option is available only when `pd` is fitted to data.
`'Discrete'`		A logical scalar to specify whether to plot the PDF or CDF of a discrete distribution object as a line plot or a stem plot, by specifying `false` or `true`, respectively. By default, it is `true` for discrete distributions and `false` for continuous distributions. When `pd` is a continuous distribution object, option is ignored.
`'Parent'`		An axes graphics object for plot. If not specified, the `plot` function plots into the current axes or creates a new axes object if one does not exist.

h = plot (…) returns a graphics handle to the plotted objects.

Example: 1

Create an Inverse Gaussian distribution with fixed parameters μ = 1 and λ = 2 and plot its PDF.

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 plot (pd)
 title ("Fixed Inverse Gaussian distribution with mu = 1 and lambda = 2")

Example: 2

Generate a data set of 100 random samples from an Inverse Gaussian distribution with parameters μ = 1 and λ = 2. Fit an Inverse Gaussian distribution to this data and plot its CDF superimposed over an empirical CDF.

 pd_fixed = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd_fixed =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 rand ("seed", 21);
 data = random (pd_fixed, 100, 1);
 pd_fitted = fitdist (data, "InverseGaussian")

pd_fitted =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 0.978108   [0.845309, 1.11091]
   lambda =   2.0383   [1.47332, 2.60328]

 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Inverse Gaussian distribution with μ = %0.2f and λ = %0.2f";
 title (sprintf (txt, pd_fitted.mu, pd_fitted.lambda))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

Use this to visualize the fitted CDF compared to the empirical CDF of the data, useful for assessing model fit.

Example: 3

Generate a data set of 200 random samples from an Inverse Gaussian distribution with parameters μ = 1 and λ = 2. Display a probability plot for the Inverse Gaussian distribution fit to the data.

 pd_fixed = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd_fixed =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 rand ("seed", 21);
 data = random (pd_fixed, 200, 1);
 pd_fitted = fitdist (data, "InverseGaussian")

pd_fitted =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 0.97456   [0.881299, 1.06782]
   lambda = 2.04403   [1.64341, 2.44465]

 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Inverse Gaussian", ...
               " distribution with μ = %0.2f and λ = %0.2f");
 title (sprintf (txt, pd_fitted.mu, pd_fitted.lambda))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

This creates a probability plot to compare the fitted distribution to the data, useful for checking if the Inverse Gaussian model is appropriate.

InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum)
InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)
InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum, setparam, 'Display', display)

[nlogL, param] = proflik (pd, pnum) returns a vector nlogL of negative loglikelihood values and a vector param of corresponding parameter values for the parameter in the position indicated by pnum. By default, proflik uses the lower and upper bounds of the 95% confidence interval and computes 100 equispaced values for the selected parameter. pd must be fitted to data.

[nlogL, param] = proflik (pd, pnum, 'Display', 'on') also plots the profile likelihood against the default range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam) defines a user-defined range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam, 'Display', 'on') also plots the profile likelihood against the user-defined range of the selected parameter.

For the Inverse Gaussian distribution, pnum = 1 selects the parameter mu and pnum = 2 selects the parameter lambda.

When opted to display the profile likelihood plot, proflik also plots the baseline loglikelihood computed at the lower bound of the 95% confidence interval and estimated maximum likelihood. The latter might not be observable if it is outside of the used-defined range of parameter values.

Example: 1

Compute and plot the profile likelihood for the shape parameter of a fitted Inverse Gaussian distribution

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 rand ("seed", 21);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "InverseGaussian")

pd_fitted =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 0.993918   [0.948235, 1.0396]
   lambda =  1.80738   [1.64896, 1.9658]

 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");

Use this to analyze the profile likelihood of the shape parameter (lambda), helping to understand the uncertainty in parameter estimates.

InverseGaussianDistribution: r = random (pd)
InverseGaussianDistribution: r = random (pd, rows)
InverseGaussianDistribution: r = random (pd, rows, cols, …)
InverseGaussianDistribution: r = random (pd, [sz])

r = random (pd) returns a random number from the distribution object pd.

When called with a single size argument, invgrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Example: 1

Generate random samples from an Inverse Gaussian distribution

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 rand ("seed", 21);
 samples = random (pd, 500, 1);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Inverse Gaussian(mu=1, lambda=2)")
 xlabel ("values")
 ylabel ("Frequency")

This generates random samples from an Inverse Gaussian distribution, useful for simulating non-negative skewed data like repair times.

InverseGaussianDistribution: s = std (pd)

s = std (pd) computes the standard deviation of the probability distribution object, pd.

Example: 1

Compute the standard deviation for an Inverse Gaussian distribution

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 std_value = std (pd)

std_value = 0.7071

Use this to calculate the standard deviation, which measures the variability in values.

InverseGaussianDistribution: t = truncate (pd, lower, upper)

t = truncate (pd, lower, upper) returns a probability distribution t, which is the probability distribution pd truncated to the specified interval with lower limit, lower, and upper limit, upper. If pd is fitted to data with fitdist, the returned probability distribution t is not fitted, does not contain any data or estimated values, and it is as it has been created with the makedist function, but it includes the truncation interval.

Example: 1

Plot the PDF of an Inverse Gaussian distribution, with parameters mu = 1 and lambda = 2, truncated at [0.5, 2] intervals. Generate 10000 random samples from this truncated distribution and superimpose a histogram scaled accordingly

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 t = truncate (pd, 0.5, 2)

t =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2
  Truncated to the interval [0.5, 2]

 rand ("seed", 21);
 data = random (t, 10000, 1);

Plot histogram and fitted PDF

 plot (t)
 hold on
 hist (data, 100, 50)
 hold off
 title ("Inverse Gaussian distribution (mu = 1, lambda = 2) truncated at [0.5, 2]")
 legend ("Truncated PDF", "Histogram")

This demonstrates truncating an Inverse Gaussian distribution to a specific range and visualizing the resulting distribution with random samples.

InverseGaussianDistribution: v = var (pd)

v = var (pd) computes the variance of the probability distribution object, pd.

Example: 1

Compute the variance for an Inverse Gaussian distribution

 pd = makedist ("InverseGaussian", "mu", 1, "lambda", 2)

pd =
  InverseGaussianDistribution

  Inverse Gaussian distribution
       mu = 1
   lambda = 2

 var_value = var (pd)

var_value = 0.5000

Use this to calculate the variance, which quantifies the spread of the values in the distribution.

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Class Definition: InverseGaussianDistribution

statistics: InverseGaussianDistribution

mu

Example: 1

Example: 2

lambda

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

cdf

InverseGaussianDistribution: p = cdf (pd, x)

InverseGaussianDistribution: p = cdf (pd, x, 'upper')

Example: 1

icdf

InverseGaussianDistribution: x = icdf (pd, p)

Example: 1

iqr

InverseGaussianDistribution: r = iqr (pd)

Example: 1

mean

InverseGaussianDistribution: m = mean (pd)

Example: 1

median

InverseGaussianDistribution: m = median (pd)

Example: 1

negloglik

InverseGaussianDistribution: nlogL = negloglik (pd)

Example: 1

paramci

Class Definition: `InverseGaussianDistribution`

`mu`

`lambda`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

`InverseGaussianDistribution: p = cdf (pd, x)`

`InverseGaussianDistribution: p = cdf (pd, x, 'upper')`

`icdf`

`InverseGaussianDistribution: x = icdf (pd, p)`

`iqr`

`InverseGaussianDistribution: r = iqr (pd)`

`mean`

`InverseGaussianDistribution: m = mean (pd)`

`median`

`InverseGaussianDistribution: m = median (pd)`

`negloglik`

`InverseGaussianDistribution: nlogL = negloglik (pd)`

`paramci`

`InverseGaussianDistribution: ci = paramci (pd)`

`InverseGaussianDistribution: ci = paramci (pd, Name, Value)`

`pdf`

`InverseGaussianDistribution: y = pdf (pd, x)`

`plot`

`InverseGaussianDistribution: plot (pd)`

`InverseGaussianDistribution: plot (pd, Name, Value)`

`InverseGaussianDistribution: h = plot (…)`

`proflik`

`InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum)`

`InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)`

`InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum, setparam)`

`InverseGaussianDistribution: [nlogL, param] = proflik (pd, pnum, setparam, 'Display', display)`

`random`

`InverseGaussianDistribution: r = random (pd)`

`InverseGaussianDistribution: r = random (pd, rows)`

`InverseGaussianDistribution: r = random (pd, rows, cols, …)`

`InverseGaussianDistribution: r = random (pd, [sz])`

`std`

`InverseGaussianDistribution: s = std (pd)`

`truncate`

`InverseGaussianDistribution: t = truncate (pd, lower, upper)`

`var`

`InverseGaussianDistribution: v = var (pd)`