### Setup Environment for Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Import necessary packages and set the random seed for reproducible results when working with distributions.

```julia
using Random, Distributions

Random.seed!(123); # Setting the seed
```

--------------------------------

### Install R Dependencies

Source: https://github.com/juliastats/distributions.jl/blob/master/test/ref/readme.md

Install R package dependencies using renv. Run this command within the R environment in the project's root directory.

```r
renv::restore()
```

--------------------------------

### Get Distribution Parameters

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Retrieve the names of the parameters used to initialize a specific distribution. This helps in understanding how to construct distribution objects.

```julia
fieldnames(Cauchy)
```

--------------------------------

### Check Distributions.jl Package Status

Source: https://github.com/juliastats/distributions.jl/blob/master/README.md

Use this command in the Julia REPL to get the current version of the Distributions package. This is useful for reporting bugs.

```julia
julia> ]status Distributions
```

--------------------------------

### Plot Density of Triweight

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of a Triweight distribution with specified parameters. This is an example of plotting a continuous distribution.

```julia
plotdensity((0, 2), Triweight, (1, 1)) # hide
```

--------------------------------

### Plot Density of Weibull Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of a Weibull distribution with specified parameters. This is an example of plotting a continuous distribution.

```julia
plotdensity((0.001, 3), Weibull, (0.5, 1)) # hide
```

--------------------------------

### Plot Density of TDist

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of a TDist with specified parameters. This is an example of plotting a continuous distribution.

```julia
plotdensity((-5, 5), TDist, (5,)) # hide
```

--------------------------------

### Instantiate Docs Project

Source: https://github.com/juliastats/distributions.jl/blob/master/README.md

Instantiate the docs project for Distributions.jl. This is a prerequisite for building the documentation locally.

```julia
julia --project=docs/
pkg> instantiate
```

--------------------------------

### Build Local Documentation

Source: https://github.com/juliastats/distributions.jl/blob/master/README.md

Build the documentation for Distributions.jl locally. This command should be run after setting up the development environment.

```julia
julia --project=docs/ docs/make.jl
```

--------------------------------

### Implement Sampler Method for Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/extends.md

Most distributions should implement a `sampler` method to enhance the efficiency of batch sampling operations.

```julia
sampler(d::Distribution)
```

--------------------------------

### Sufficient Statistics

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/fit.md

This section describes how to compute sufficient statistics from a dataset and then use them to fit a distribution.

```APIDOC
## `suffstats(D, x)` and `suffstats(D, x, w)`

### Description
Computes the sufficient statistics for a distribution of type `D` from a dataset `x`, optionally with weights `w`.

### Method
`suffstats`

### Parameters
#### Path Parameters
- **D** (Type) - Required - The type of the distribution.
- **x** (Array) - Required - The dataset (array of samples).
- **w** (Array, Optional) - The weights for each sample in `x`.

### Response
#### Success Response (Sufficient Statistics object)
- Returns an object capturing the sufficient statistics.

### Response Example
```julia
# Example for Normal distribution (mean and variance)
ss = suffstats(Normal, [1.0, 2.0, 3.0, 4.0, 5.0])
```

## `fit_mle(D, ss)`

### Description
Performs maximum likelihood estimation for a distribution of type `D` based on pre-computed sufficient statistics `ss`.

### Method
`fit_mle`

### Parameters
#### Path Parameters
- **D** (Type) - Required - The type of the distribution.
- **ss** (Sufficient Statistics object) - Required - The sufficient statistics object.

### Response
#### Success Response (Distribution object)
- Returns an instance of the fitted distribution `D`.

### Response Example
```julia
# Assuming 'ss' is a sufficient statistics object for Normal distribution
d = fit_mle(Normal, ss)
```
```

--------------------------------

### NamedTuple-variate Product Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/product.md

Constructs a product distribution from a NamedTuple of distributions.

```APIDOC
## NamedTuple-variate Product Distributions

### Description
Constructs a product distribution from a NamedTuple of distributions.

### Method
- `Distributions.product_distribution(::NamedTuple{<:Any,<:Tuple{Distribution,Vararg{Distribution}}})

### Type
- `Distributions.ProductNamedTupleDistribution`
```

--------------------------------

### Common Interface for Matrix Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/matrix.md

Lists the common methods implemented by all matrix-variate distributions, such as size, length, rank, mean, variance, covariance, and probability density functions.

```julia
size(::MatrixDistribution)
length(::MatrixDistribution)
Distributions.rank(::MatrixDistribution)
mean(::MatrixDistribution)
var(::MatrixDistribution)
cov(::MatrixDistribution)
pdf(d::MatrixDistribution, x::AbstractMatrix{<:Real})
logpdf(d::MatrixDistribution, x::AbstractMatrix{<:Real})
Distributions._rand!(::AbstractRNG, ::MatrixDistribution, A::AbstractMatrix)
```

--------------------------------

### Internal Methods for Creating Matrix-variate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/matrix.md

Provides access to internal methods, primarily intended for developers who wish to create their own custom matrix-variate distributions.

```APIDOC
## Internal Methods (for creating your own matrix-variate distributions)

```julia
Distributions._logpdf(d::MatrixDistribution, x::AbstractMatrix{<:Real})
```
```

--------------------------------

### Available Matrix-variate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/matrix.md

Lists the specific matrix-variate distributions available in the package, including MatrixNormal, Wishart, InverseWishart, MatrixTDist, MatrixBeta, MatrixFDist, and LKJ.

```julia
MatrixNormal
Wishart
InverseWishart
MatrixTDist
MatrixBeta
MatrixFDist
LKJ
```

--------------------------------

### Develop Distributions.jl Locally

Source: https://github.com/juliastats/distributions.jl/blob/master/README.md

Develop the Distributions.jl package locally. This command should be run after instantiating the docs project.

```julia
pkg> dev .
```

--------------------------------

### Define Other Distribution Types

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Illustrates how to define various types of distributions including discrete, continuous, univariate, multivariate, and matrix-variate distributions.

```julia
Binomial(n, p) # Discrete univariate
```

```julia
Cauchy(u, b)  # Continuous univariate
```

```julia
Multinomial(n, p) # Discrete multivariate
```

```julia
Wishart(nu, S) # Continuous matrix-variate
```

--------------------------------

### Implement Univariate Sampler

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/extends.md

To implement a univariate sampler, define a subtype of `Sampleable{Univariate,S}` and provide a `rand` method for generating a single sample.

```julia
function rand(rng::AbstractRNG, s::Spl)
    # ... generate a single sample from s
end
```

--------------------------------

### Construct Mixture of Normal Distributions with Prior

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/mixture.md

Constructs a mixture of three univariate normal distributions with specified prior probabilities. Ensure the component distributions and prior probabilities are provided in the correct order.

```julia
MixtureModel(Normal[Normal(-2.0, 1.2), Normal(0.0, 1.0), Normal(3.0, 2.5)], [0.2, 0.5, 0.3])
```

--------------------------------

### Implement Matrix-variate Sampler

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/extends.md

To implement a matrix-variate sampler, define a subtype of `Sampleable{Multivariate,S}` and implement `size` and `_rand!` methods for generating matrix samples.

```julia
Base.size(s::Spl) = ... # the size of each matrix sample

function _rand!(rng::AbstractRNG, s::Spl, x::DenseMatrix{T}) where T<:Real
    # ... generate a single matrix sample to x
end
```

--------------------------------

### MvLogNormal Specific Methods

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Lists methods specific to the `MvLogNormal` distribution, including functions to retrieve its location, scale, median, and mode. It also provides functions to calculate the location and scale parameters from mean and covariance, or median/mode, which is useful for parameter transformations.

```julia
location(::MvLogNormal)
scale(::MvLogNormal)
median(::MvLogNormal)
mode(::MvLogNormal)
```

```julia
location{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix)
location!{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix,μ::AbstractVector)
scale{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix)
scale!{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix,Σ::AbstractMatrix)
params{D<:Distributions.AbstractMvLogNormal}(::Type{D},m::AbstractVector,S::AbstractMatrix)
```

--------------------------------

### Retrieve Distribution Parameters

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Provides methods to retrieve various parameters of univariate distributions. Not all methods are defined for all distributions.

```julia
params(::UnivariateDistribution)
scale(::UnivariateDistribution)
location(::UnivariateDistribution)
shape(::UnivariateDistribution)
rate(::UnivariateDistribution)
ncategoriess(::UnivariateDistribution)
ntrials(::UnivariateDistribution)
dof(::UnivariateDistribution)
```

--------------------------------

### Uniform

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Documentation for the Uniform distribution.

```APIDOC
## Uniform

### Description
Documentation for the Uniform distribution.

### Example
```julia
plotdensity((-0.5, 1.5), Uniform, (0, 1); ylim=(0, 1.5))
```
```

--------------------------------

### Efficient Batch Sampling for Multivariate Samplers

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/extends.md

If a more efficient method exists for generating multiple vector samples in a batch, provide an optimized `_rand!` method that accepts a `DenseMatrix`.

```julia
function _rand!(rng::AbstractRNG, s::Spl, A::DenseMatrix{T}) where T<:Real
    # ... generate multiple vector samples in batch
end
```

--------------------------------

### Construct Mixture with Parameter Tuples and Prior

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/mixture.md

Constructs a mixture model using tuples of parameters for the component distributions and an explicit prior probability vector. This syntax simplifies component definition when they share the same type.

```julia
MixtureModel(Normal, [(-2.0, 1.2), (0.0, 1.0), (3.0, 2.5)], [0.2, 0.5, 0.3])
```

--------------------------------

### Additional Methods for AbstractMvNormal

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Extended methods available for distributions inheriting from AbstractMvNormal.

```APIDOC
## Addition Methods

### AbstractMvNormal

In addition to the methods listed in the common interface above, we also provide the following methods for all multivariate distributions under the base type `AbstractMvNormal`:

### `invcov(::Distributions.AbstractMvNormal)`

Computes the inverse of the covariance matrix.

### `logdetcov(::Distributions.AbstractMvNormal)`

Computes the logarithm of the determinant of the covariance matrix.

### `sqmahal(::Distributions.AbstractMvNormal, ::AbstractArray)`

Computes the squared Mahalanobis distance.

### `rand(::AbstractRNG, ::Distributions.AbstractMvNormal)`

Generates a random sample from the distribution.

### `minimum(::Distributions.AbstractMvNormal)`

Returns the minimum value in the support of the distribution.

### `maximum(::Distributions.AbstractMvNormal)`

Returns the maximum value in the support of the distribution.

### `extrema(::Distributions.AbstractMvNormal)`

Returns the minimum and maximum values in the support of the distribution.
```

--------------------------------

### AbstractMvNormal Additional Methods

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Provides additional methods specific to distributions inheriting from `AbstractMvNormal`. These include calculations related to the covariance matrix such as its inverse and log-determinant, squared Mahalanobis distance, and sampling. It also includes methods for finding the minimum, maximum, and extrema of the distribution's support.

```julia
invcov(::Distributions.AbstractMvNormal)
logdetcov(::Distributions.AbstractMvNormal)
sqmahal(::Distributions.AbstractMvNormal, ::AbstractArray)
rand(::AbstractRNG, ::Distributions.AbstractMvNormal)
minimum(::Distributions.AbstractMvNormal)
maximum(::Distributions.AbstractMvNormal)
extrema(::Distributions.AbstractMvNormal)
```

--------------------------------

### Common Interface Methods

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/mixture.md

Provides access to the components, their prior probabilities, and the type of the component distributions for any mixture model.

```APIDOC
## Common Interface

```julia
components(::AbstractMixtureModel)
probs(::AbstractMixtureModel)
Distributions.component_type(::AbstractMixtureModel)
```

### Description

These methods allow you to retrieve the individual component distributions, their associated prior probabilities, and the type of the component distributions from a mixture model.

### Examples

```julia
# Assuming `mm` is an instance of MixtureModel
comps = components(mm)
priors = probs(mm)
comp_type = Distributions.component_type(mm)
```
```

--------------------------------

### Multivariate Product Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/product.md

Constructs a product distribution from an array of distributions or a vector of Normal distributions.

```APIDOC
## Multivariate Product Distributions

### Description
Constructs a product distribution from an array of distributions or a vector of Normal distributions.

### Methods
- `Distributions.product_distribution(::AbstractArray{<:Distribution{<:ArrayLikeVariate}})
- `Distributions.product_distribution(::AbstractVector{<:Normal})

### Types
- `Distributions.ProductDistribution`
- `Distributions.Product`
```

--------------------------------

### Retrieve Success and Failure Probabilities

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Defines methods for success and failure probabilities, applicable to discrete distributions where these concepts are meaningful.

```julia
succprob(::DiscreteUnivariateDistribution)
failprob(::DiscreteUnivariateDistribution)
```

--------------------------------

### Create and Inspect Standard Normal Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Create a standard-normal distribution object and query its mean. This is useful for understanding the basic properties of a distribution.

```julia
d = Normal()
mean(d)
```

--------------------------------

### Internal Method for Creating Matrix-variate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/matrix.md

Details the internal method `_logpdf` used for creating custom matrix-variate distributions.

```julia
Distributions._logpdf(d::MatrixDistribution, x::AbstractMatrix{<:Real})
```

--------------------------------

### Specific Matrix-variate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/matrix.md

This section lists the available specific matrix-variate distributions implemented in the package.

```APIDOC
## Distributions

```julia
MatrixNormal
Wishart
InverseWishart
MatrixTDist
MatrixBeta
MatrixFDist
LKJ
```
```

--------------------------------

### Multivariate Distribution Common Interface

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Lists common methods for multivariate distributions, including computation of statistics like length, size, mean, variance, standard deviation, covariance, correlation, and entropy. These methods provide a consistent way to interact with any multivariate distribution.

```julia
length(::MultivariateDistribution)
size(::MultivariateDistribution)
eltype(::Type{MultivariateDistribution})
mean(::MultivariateDistribution)
var(::MultivariateDistribution)
std(::MultivariateDistribution)
cov(::MultivariateDistribution)
cor(::MultivariateDistribution)
entropy(::MultivariateDistribution)
entropy(::MultivariateDistribution, ::Real)
```

--------------------------------

### Common Interface for Matrix Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/matrix.md

All matrix-variate distributions implement a common set of methods for size, length, rank, mean, variance, covariance, probability density function (pdf), log probability density function (logpdf), and random number generation.

```APIDOC
## Common Interface

All distributions implement the same set of methods:

```julia
size(::MatrixDistribution)
length(::MatrixDistribution)
Distributions.rank(::MatrixDistribution)
mean(::MatrixDistribution)
var(::MatrixDistribution)
cov(::MatrixDistribution)
pdf(d::MatrixDistribution, x::AbstractMatrix{<:Real})
logpdf(d::MatrixDistribution, x::AbstractMatrix{<:Real})
Distributions._rand!(::AbstractRNG, ::MatrixDistribution, A::AbstractMatrix)
```
```

--------------------------------

### Evaluate Univariate Distribution Probabilities

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Provides functions for probability evaluation, including checking if a value is within the support, calculating PDF, log-PDF, CDF, log-CDF, and quantiles.

```julia
insupport(::UnivariateDistribution, x::Any)
pdf(::UnivariateDistribution, ::Real)
logpdf(::UnivariateDistribution, ::Real)
gradlogpdf(::ContinuousUnivariateDistribution, ::Real)
loglikelihood(::UnivariateDistribution, ::AbstractArray)
cdf(::UnivariateDistribution, ::Real)
logcdf(::UnivariateDistribution, ::Real)
logdiffcdf(::UnivariateDistribution, ::Real, ::Real)
ccdf(::UnivariateDistribution, ::Real)
logccdf(::UnivariateDistribution, ::Real)
quantile(::UnivariateDistribution, ::Real)
cquantile(::UnivariateDistribution, ::Real)
invlogcdf(::UnivariateDistribution, ::Real)
invlogccdf(::UnivariateDistribution, ::Real)
```

--------------------------------

### Weibull

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Documentation for the Weibull distribution.

```APIDOC
## Weibull

### Description
Documentation for the Weibull distribution.

### Example
```julia
plotdensity((0.001, 3), Weibull, (0.5, 1))
```
```

--------------------------------

### Generic Methods for Univariate and Multivariate Mixtures

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/mixture.md

Provides common statistical functions like mean, variance, length, probability density, log-probability density, and random sampling for univariate and multivariate mixture models.

```APIDOC
## Generic Methods

```julia
mean(::AbstractMixtureModel)
var(::UnivariateMixture)
length(::MultivariateMixture)
pdf(::AbstractMixtureModel, x)
logpdf(::AbstractMixtureModel, x)
rand(::AbstractMixtureModel)
rand!(::AbstractMixtureModel, ::AbstractArray)
```

### Description

These generic methods are available for univariate and multivariate mixture models, providing functionalities to calculate the mean, variance, length, probability density, log-probability density, and to generate random samples from the mixture distribution.

### Examples

```julia
# Assuming `mm` is an instance of UnivariateMixture or MultivariateMixture
model_mean = mean(mm)
model_var = var(mm) # For UnivariateMixture
num_components = length(mm) # For MultivariateMixture
probability = pdf(mm, value)
log_probability = logpdf(mm, value)
random_sample = rand(mm)
```
```

--------------------------------

### Sampling from Multivariate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Methods for generating random samples from multivariate distributions.

```APIDOC
## Sampling

### `rand(rng::AbstractRNG, ::MultivariateDistribution)`

Generates a random sample from the distribution using the provided random number generator.

### `rand!(rng::AbstractRNG, d::MultivariateDistribution, x::AbstractArray)`

Fills a pre-allocated array with random samples from the distribution.
```

--------------------------------

### Generate Reference Data

Source: https://github.com/juliastats/distributions.jl/blob/master/test/ref/readme.md

Execute the R script to generate reference data files. Navigate to the directory containing gendref.R in your terminal before running.

```shell
Rscript gendref.R
```

--------------------------------

### Implement Multivariate Sampler

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/extends.md

For multivariate samplers, define a subtype of `Sampleable{Multivariate,S}` and implement `length` and `_rand!` methods for generating vector samples.

```julia
Base.length(s::Spl) = ... # return the length of each sample

function _rand!(rng::AbstractRNG, s::Spl, x::AbstractVector{T}) where T<:Real
    # ... generate a single vector sample to x
end
```

--------------------------------

### Construct Mixture with Parameter Tuples (Implicit Prior)

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/mixture.md

Constructs a mixture model using parameter tuples for components, omitting the prior vector to imply equal prior probabilities for all components. This is a concise way to define mixtures with uniform priors.

```julia
MixtureModel(Normal, [(-2.0, 1.2), (0.0, 1.0), (3.0, 2.5)])
```

--------------------------------

### Additional Methods for MvLogNormal

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Specific methods for the MvLogNormal distribution.

```APIDOC
### MvLogNormal

In addition to the methods listed in the common interface above, we also provide the following methods:

### `location(::MvLogNormal)`

Returns the location vector of the log-normal distribution.

### `scale(::MvLogNormal)`

Returns the scale matrix of the log-normal distribution.

### `median(::MvLogNormal)`

Computes the median of the log-normal distribution.

### `mode(::MvLogNormal)`

Computes the mode of the log-normal distribution.

It can be necessary to calculate the parameters of the lognormal (location vector and scale matrix) from a given covariance and mean, median or mode. To that end, the following functions are provided.

### `location{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix)`

Calculates the location parameter for a log-normal distribution type.

### `location!{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix,μ::AbstractVector)`

In-place version of `location`.

### `scale{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix)`

Calculates the scale parameter for a log-normal distribution type.

### `scale!{D<:Distributions.AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix,Σ::AbstractMatrix)`

In-place version of `scale`.

### `params{D<:Distributions.AbstractMvLogNormal}(::Type{D},m::AbstractVector,S::AbstractMatrix)`

Calculates the parameters (location and scale) for a log-normal distribution type.
```

--------------------------------

### Detect Method Ambiguities

Source: https://github.com/juliastats/distributions.jl/blob/master/README.md

Run this command at the end of the tests to check for method ambiguities introduced by your modifications. This helps maintain code quality.

```julia
Test.detect_ambiguities(Distributions)
```

--------------------------------

### Plot Density of Continuous Univariate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

A helper function to plot the probability density function of continuous univariate distributions. Requires Distributions and GR packages.

```julia
using Distributions, GR

GR.inline("svg")

function plotdensity(
    (xmin, xmax),
    dist::ContinuousUnivariateDistribution;
    npoints=299,
    title="",
    kwargs...
)
    figure(;
        title=title,
        xlabel="x",
        ylabel="density",
        grid=false,
        backgroundcolor=0, # white instead of transparent background for dark Documenter scheme
        font="Helvetica_Regular", # work around https://github.com/JuliaPlots/Plots.jl/issues/2596
        linewidth=2.0, # thick lines
        kwargs...
    )
    return plot(range(xmin, xmax; length=npoints), Base.Fix1(pdf, dist))
end
```

--------------------------------

### Sampling (Random number generation)

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Functions for generating random numbers from univariate distributions.

```APIDOC
## Sampling (Random number generation)

```julia
rand(::AbstractRNG, ::UnivariateDistribution)
rand!(::AbstractRNG, ::UnivariateDistribution, ::AbstractArray)
```
```

--------------------------------

### TriangularDist

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Documentation for the Triangular distribution.

```APIDOC
## TriangularDist

### Description
Documentation for the Triangular distribution.

### Example
```julia
plotdensity((-0.5, 2), TriangularDist, (0, 1.5, 0.5); npoints=6)
```
```

--------------------------------

### Create Censored Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/censored.md

Use the `censored` function to create a censored version of a given distribution. This is the recommended way to construct censored distributions.

```julia
censored
```

--------------------------------

### Multivariate Distribution Sampling

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Details methods for sampling from multivariate distributions. Includes functions for generating random variates and efficiently filling pre-allocated arrays with samples. These are essential for Monte Carlo simulations and statistical analysis.

```julia
rand(rng::AbstractRNG, ::MultivariateDistribution)
rand!(rng::AbstractRNG, d::MultivariateDistribution, x::AbstractArray)
```

--------------------------------

### Plot Density for BetaPrime Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Beta Prime distribution. Requires specifying the range and shape parameters.

```julia
plotdensity((0, 1), BetaPrime, (1, 2)) # hide
```

--------------------------------

### Multivariate Distribution Probability Evaluation

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Covers methods for probability evaluation in multivariate distributions, such as checking if a point is within the support, calculating the probability density function (pdf), and the log-probability density function (logpdf). It is generally advisable to use log scale for probability computations due to potential numerical issues with very small or large pdf values.

```julia
insupport(::MultivariateDistribution, ::AbstractArray)
pdf(::MultivariateDistribution, ::AbstractArray)
logpdf(::MultivariateDistribution, ::AbstractArray)
loglikelihood(::MultivariateDistribution, ::AbstractVector{<:Real})
```

--------------------------------

### Construct Mixture of Normal Distributions (Implicit Prior)

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/mixture.md

Constructs a mixture of normal distributions where the prior probabilities are assumed to be equal, as the prior vector is omitted. This is suitable when all components have the same weight.

```julia
MixtureModel(Normal[Normal(-2.0, 1.2), Normal(0.0, 1.0), Normal(3.0, 2.5)])
```

--------------------------------

### MixtureModel Constructors

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/mixture.md

Constructs a mixture model by providing a vector of component distributions and an optional prior probability vector. If the prior vector is omitted, components are assumed to have equal prior probabilities.

```APIDOC
## Constructors

```julia
MixtureModel(components::Vector{<:Distribution}, prior::Categorical)
MixtureModel(components::Vector{<:Distribution})
MixtureModel{VF,VS,Component}(components::Vector{Component}, prior::Categorical)
MixtureModel{VF,VS,Component}(components::Vector{Component})
```

### Description

Constructs a mixture model from a vector of component distributions and an optional prior distribution. If the prior is omitted, it is assumed to be a uniform categorical distribution over the components.

### Examples

```julia
# Mixture of three normal distributions with specified prior probabilities
MixtureModel(Normal[Normal(-2.0, 1.2), Normal(0.0, 1.0), Normal(3.0, 2.5)], Categorical([0.2, 0.5, 0.3]))

# Mixture of three normal distributions with equal prior probabilities
MixtureModel(Normal[Normal(-2.0, 1.2), Normal(0.0, 1.0), Normal(3.0, 2.5)])

# Simplified syntax when component types are known and priors are equal
MixtureModel(Normal, [(-2.0, 1.2), (0.0, 1.0), (3.0, 2.5)])

# Gaussian mixture with unit variance components
MixtureModel(map(u -> Normal(u, 1.0), [-2.0, 0.0, 3.0]))
```
```

--------------------------------

### Sample from a Normal Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Generate random samples from a specified normal distribution. This is useful for simulations or creating test data.

```julia
x = rand(d, 100)
```

```julia
rand(Normal(1, 2), 100)
```

--------------------------------

### VonMises

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Documentation for the Von Mises distribution.

```APIDOC
## VonMises

### Description
Documentation for the Von Mises distribution.

### Example
```julia
plotdensity((-π, π), VonMises, (0.5,); xlim=(-π, π), xticks=(π/5, 5), xticklabels=x -> x ≈ -π ? "-π" : (x ≈ π ? "π" : "0"))
```
```

--------------------------------

### Plot Density for Johnson SU Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Johnson SU distribution. Requires specifying the range and parameters.

```julia
plotdensity((-20, 20), JohnsonSU, (0.0, 1.0, 0.0, 1.0)) # hide
```

--------------------------------

### Plot Density for Generalized Extreme Value Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Generalized Extreme Value distribution. Requires specifying the range and shape/scale/location parameters.

```julia
plotdensity((0, 30), GeneralizedExtremeValue, (0, 1, 1)) # hide
```

--------------------------------

### Plot Density for LogUniform Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Log-Uniform distribution. Requires specifying the range and parameters.

```julia
plotdensity((0, 11), LogUniform, (1, 10)) # hide
```

--------------------------------

### Plot Density for Skewed Exponential Power Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Skewed Exponential Power distribution. Requires specifying the range and parameters.

```julia
plotdensity((-8, 5), SkewedExponentialPower, (0, 1, 0.7, 0.7)) # hide
```

--------------------------------

### Parameter Retrieval

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Methods for retrieving parameters of univariate distributions. Some methods are distribution-specific.

```APIDOC
## Parameter Retrieval

**Note:** `params` are defined for all univariate distributions, while other parameter retrieval methods are only defined for those distributions for which these parameters make sense. See below for details.

```julia
params(::UnivariateDistribution)
scale(::UnivariateDistribution)
location(::UnivariateDistribution)
shape(::UnivariateDistribution)
rate(::UnivariateDistribution)
ncategoriess(::UnivariateDistribution)
ntrials(::UnivariateDistribution)
dof(::UnivariateDistribution)
```

For distributions for which success and failure have a meaning, the following methods are defined:

```julia
succprob(::DiscreteUnivariateDistribution)
failprob(::DiscreteUnivariateDistribution)
```
```

--------------------------------

### Distribution Type Aliases

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/types.md

Introduces type aliases for common combinations of variate forms and value supports to simplify practical usage of distributions.

```julia
const UnivariateDistribution{S<:ValueSupport}   = Distribution{Univariate,S}
const MultivariateDistribution{S<:ValueSupport} = Distribution{Multivariate,S}
const MatrixDistribution{S<:ValueSupport}       = Distribution{Matrixvariate,S}
const NonMatrixDistribution = Union{UnivariateDistribution, MultivariateDistribution}

const DiscreteDistribution{F<:VariateForm}   = Distribution{F,Discrete}
const ContinuousDistribution{F<:VariateForm} = Distribution{F,Continuous}

const DiscreteUnivariateDistribution     = Distribution{Univariate,    Discrete}
const ContinuousUnivariateDistribution   = Distribution{Univariate,    Continuous}
const DiscreteMultivariateDistribution   = Distribution{Multivariate,  Discrete}
const ContinuousMultivariateDistribution = Distribution{Multivariate,  Continuous}
const DiscreteMatrixDistribution         = Distribution{Matrixvariate, Discrete}
const ContinuousMatrixDistribution       = Distribution{Matrixvariate, Continuous}
```

--------------------------------

### Plot Density for Kolmogorov Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Kolmogorov distribution. Requires specifying the range.

```julia
plotdensity((0, 2), Kolmogorov) # hide
```

--------------------------------

### Internal Method for Log-PDF Calculation

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Defines the internal method `_logpdf` for calculating the log-probability density function of a multivariate distribution. This is primarily intended for users who are creating their own custom multivariate distribution types.

```julia
Distributions._logpdf(d::MultivariateDistribution, x::AbstractArray)
```

--------------------------------

### Plot Density for Frechet Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Frechet distribution. Requires specifying the range and shape parameters.

```julia
plotdensity((0, 20), Frechet, (1, 1)) # hide
```

--------------------------------

### Discrete Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Documentation for various discrete probability distributions.

```APIDOC
## Discrete Distributions

### Bernoulli
### BernoulliLogit
### BetaBinomial
### Binomial
### Categorical
### Dirac
### DiscreteUniform
### DiscreteNonParametric
### Geometric
### Hypergeometric
### NegativeBinomial
### Poisson
### PoissonBinomial
### Skellam
### Soliton
```

--------------------------------

### Triweight

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Documentation for the Triweight distribution.

```APIDOC
## Triweight

### Description
Documentation for the Triweight distribution.

### Example
```julia
plotdensity((0, 2), Triweight, (1, 1))
```
```

--------------------------------

### Probability Evaluation for Multivariate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

Methods for evaluating probability density functions and related quantities.

```APIDOC
## Probability Evaluation

### `insupport(::MultivariateDistribution, ::AbstractArray)`

Checks if a given vector is within the support of the distribution.

### `pdf(::MultivariateDistribution, ::AbstractArray)`

Computes the probability density function (PDF) at a given point.

### `logpdf(::MultivariateDistribution, ::AbstractArray)`

Computes the logarithm of the probability density function (log-PDF) at a given point.

### `loglikelihood(::MultivariateDistribution, ::AbstractVector{<:Real})`

Computes the log-likelihood of a single observation.

**Note:** For multivariate distributions, the pdf value is usually very small or large, and therefore direct evaluation of the pdf may cause numerical problems. It is generally advisable to perform probability computation in log scale.
```

--------------------------------

### Plot Density for Erlang Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Erlang distribution. Requires specifying the range and shape/scale parameters.

```julia
plotdensity((0, 8), Erlang, (7, 0.5)) # hide
```

--------------------------------

### Plot Density for Beta Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Beta distribution. Requires specifying the range and shape parameters.

```julia
plotdensity((0, 1), Beta, (2, 2)) # hide
```

--------------------------------

### Fit Binomial Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/fit.md

Fit a Binomial distribution using the number of trials `n` and the observed data `x`. Supports weighted data.

```julia
fit_mle(Binomial, n, x)
```

```julia
fit_mle(Binomial, n, x, w)
```

--------------------------------

### Compute Univariate Distribution Statistics

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Lists common statistical computations available for univariate distributions, including moments, measures of spread, and shape.

```julia
maximum(::UnivariateDistribution)
minimum(::UnivariateDistribution)
extrema(::UnivariateDistribution)
mean(::UnivariateDistribution)
var(::UnivariateDistribution)
std(::UnivariateDistribution)
median(::UnivariateDistribution)
modes(::UnivariateDistribution)
mode(::UnivariateDistribution)
skewness(::UnivariateDistribution)
kurtosis(::UnivariateDistribution)
kurtosis(::Distribution, ::Bool)
isplatykurtic(::UnivariateDistribution)
isleptokurtic(::UnivariateDistribution)
ismesokurtic(::UnivariateDistribution)
entropy(::UnivariateDistribution)
entropy(::UnivariateDistribution, ::Real)
mgf(::UnivariateDistribution, ::Any)
cgf(::UnivariateDistribution, ::Any)
cf(::UnivariateDistribution, ::Any)
pdfsquaredL2norm
```

--------------------------------

### Plot Density of TriangularDist

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of a TriangularDist with specified parameters and limits. Uses 6 equally spaced points for plotting.

```julia
plotdensity((-0.5, 2), TriangularDist, (0, 1.5, 0.5); npoints=6) # hide
```

--------------------------------

### Fit Distribution from Sufficient Statistics

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/fit.md

Perform maximum likelihood estimation for a distribution `D` using pre-computed sufficient statistics `ss`.

```julia
d = fit_mle(D, ss)
```

--------------------------------

### Plot Density for Biweight Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Biweight distribution. Requires specifying the range and shape parameters.

```julia
plotdensity((-1, 3), Biweight, (1, 2)) # hide
```

--------------------------------

### Plot Density for Cauchy Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Cauchy distribution. Requires specifying the range and location-scale parameters.

```julia
plotdensity((-12, 5), Cauchy, (-2, 1)) # hide
```

--------------------------------

### Plot Density for Generalized Pareto Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Generalized Pareto distribution. Requires specifying the range and shape/scale/location parameters.

```julia
plotdensity((0, 20), GeneralizedPareto, (0, 1, 1)) # hide
```

--------------------------------

### Create Truncated Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Define a truncated version of an existing univariate distribution. This is useful when you need to constrain the range of a distribution.

```julia
truncated(Normal(mu, sigma), l, u)
```

--------------------------------

### Plot Density for Normal Inverse Gaussian Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Normal Inverse Gaussian distribution. Requires specifying the range and parameters.

```julia
plotdensity((-2, 2), NormalInverseGaussian, (0, 0.5, 0.2, 0.1)) # hide
```

--------------------------------

### Maximum Likelihood Estimation (MLE)

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/fit.md

The `fit_mle` function is specifically for maximum likelihood estimation. It can be used with or without weights.

```APIDOC
## `fit_mle(D, x)` and `fit_mle(D, x, w)`

### Description
Performs maximum likelihood estimation to fit a distribution of type `D` to a dataset `x`, optionally with weights `w`.

### Method
`fit_mle`

### Parameters
#### Path Parameters
- **D** (Type) - Required - The type of the distribution to fit.
- **x** (Array) - Required - The dataset (array of samples).
- **w** (Array, Optional) - The weights for each sample in `x`.

### Request Example
```julia
# Without weights
d_unweighted = fit_mle(Normal, [1.0, 2.0, 3.0, 4.0, 5.0])

# With weights
weights = [0.5, 1.0, 1.5, 2.0, 2.5]
d_weighted = fit_mle(Normal, [1.0, 2.0, 3.0, 4.0, 5.0], weights)
```

### Response
#### Success Response (Distribution object)
- Returns an instance of the fitted distribution `D`.

### Response Example
```julia
Normal{Float64}(μ=3.0, σ=1.4142135623730951)
```
```

--------------------------------

### Computation of Statistics for Multivariate Distributions

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/multivariate.md

These methods provide common statistical computations for multivariate distributions.

```APIDOC
## Computation of Statistics

### `length(::MultivariateDistribution)`

Returns the number of dimensions of the distribution.

### `size(::MultivariateDistribution)`

Returns the size of the distribution (number of dimensions).

### `eltype(::Type{MultivariateDistribution})`

Returns the element type of the samples from the distribution.

### `mean(::MultivariateDistribution)`

Computes the mean vector of the distribution.

### `var(::MultivariateDistribution)`

Computes the variance vector of the distribution.

### `std(::MultivariateDistribution)`

Computes the standard deviation vector of the distribution.

### `cov(::MultivariateDistribution)`

Computes the covariance matrix of the distribution.

### `cor(::MultivariateDistribution)`

Computes the correlation matrix of the distribution.

### `entropy(::MultivariateDistribution)`

Computes the differential entropy of the distribution.

### `entropy(::MultivariateDistribution, ::Real)`

Computes the conditional entropy of the distribution with respect to a given value.
```

--------------------------------

### Maximum Likelihood Estimation Synopsis

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/fit.md

These functions are used for maximum likelihood estimation of distribution parameters.

```julia
fit(D, x)
```

```julia
fit(D, x, w)
```

```julia
fit_mle(D, x)
```

```julia
fit_mle(D, x, w)
```

--------------------------------

### Plot Density for Cosine Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Cosine distribution. Requires specifying the range and parameters.

```julia
plotdensity((-1, 1), Cosine, (0, 1)) # hide
```

--------------------------------

### Plot Density for Pareto Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Pareto distribution. Requires specifying the range and shape/scale parameters.

```julia
plotdensity((1, 8), Pareto, (1, 1)) # hide
```

--------------------------------

### Plot Density for Logit Normal Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Logit Normal distribution. Requires specifying the range and location-scale parameters.

```julia
plotdensity((0, 1), LogitNormal, (0, 1)) # hide
```

--------------------------------

### Plot Density for Gamma Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Gamma distribution. Requires specifying the range and shape/scale parameters.

```julia
plotdensity((0, 18), Gamma, (7.5, 1)) # hide
```

--------------------------------

### Calculate Quantiles for Standard Normal Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Compute quantiles for a given distribution. This is useful for understanding percentiles and probability thresholds.

```julia
quantile.(Normal(), [0.5, 0.95])
```

--------------------------------

### Plot Density for Inverse Gaussian Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Inverse Gaussian distribution. Requires specifying the range and mean/dispersion parameters.

```julia
plotdensity((0, 5), InverseGaussian, (1, 1)) # hide
```

--------------------------------

### Plot Density for PGeneralizedGaussian Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the P-Generalized Gaussian distribution. Requires specifying the range and parameter.

```julia
plotdensity((0, 20), PGeneralizedGaussian, (0.2)) # hide
```

--------------------------------

### Plot Density for Laplace Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Laplace distribution. Requires specifying the range and location-scale parameters.

```julia
plotdensity((-20, 20), Laplace, (0, 4)) # hide
```

--------------------------------

### Plot Density for Arcsine Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Arcsine distribution. Requires specifying the range and distribution parameters.

```julia
plotdensity((0.001, 0.999), Arcsine, (0, 1)) # hide
```

--------------------------------

### Plot Density of Uniform Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of a Uniform distribution with specified parameters and limits. Includes setting the y-axis limits.

```julia
plotdensity((-0.5, 1.5), Uniform, (0, 1); ylim=(0, 1.5)) # hide
```

--------------------------------

### Plot Density for Rayleigh Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Rayleigh distribution. Requires specifying the range and scale parameter.

```julia
plotdensity((0, 2), Rayleigh, (0.5)) # hide
```

--------------------------------

### Plot Density for Skew Normal Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Skew Normal distribution. Requires specifying the range and parameters.

```julia
plotdensity((-4, 4), SkewNormal, (0, 1, -1)) # hide
```

--------------------------------

### Plot Density for F Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the F distribution. Requires specifying the range and degrees of freedom.

```julia
plotdensity((0, 10), FDist, (10, 1)) # hide
```

--------------------------------

### Probability Evaluation

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Methods for evaluating probabilities, including PDF, CDF, and related logarithmic and inverse functions.

```APIDOC
## Probability Evaluation

```julia
insupport(::UnivariateDistribution, x::Any)
pdf(::UnivariateDistribution, ::Real)
logpdf(::UnivariateDistribution, ::Real)
gradlogpdf(::ContinuousUnivariateDistribution, ::Real)
loglikelihood(::UnivariateDistribution, ::AbstractArray)
cdf(::UnivariateDistribution, ::Real)
logcdf(::UnivariateDistribution, ::Real)
logdiffcdf(::UnivariateDistribution, ::Real, ::Real)
ccdf(::UnivariateDistribution, ::Real)
logccdf(::UnivariateDistribution, ::Real)
quantile(::UnivariateDistribution, ::Real)
cquantile(::UnivariateDistribution, ::Real)
invlogcdf(::UnivariateDistribution, ::Real)
invlogccdf(::UnivariateDistribution, ::Real)
```
```

--------------------------------

### Plot Density of VonMises Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of a VonMises distribution with specified parameters. Includes setting x-axis limits and custom tick labels for π.

```julia
plotdensity((-π, π), VonMises, (0.5,); xlim=(-π, π), xticks=(π/5, 5), xticklabels=x -> x ≈ -π ? "-π" : (x ≈ π ? "π" : "0")) # hide
```

--------------------------------

### Fit Distribution to Data

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/fit.md

Fit a distribution of type `D` to a dataset `x`. This typically uses maximum likelihood estimation.

```julia
d = fit(D, x)
```

--------------------------------

### Plot Density for Inverse Gamma Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Inverse Gamma distribution. Requires specifying the range and shape/scale parameters.

```julia
plotdensity((0.001, 1), InverseGamma, (3, 0.5)) # hide
```

--------------------------------

### Plot Density for NormalCanon Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the NormalCanon distribution. Requires specifying the range and parameters.

```julia
plotdensity((-4, 4), NormalCanon, (0, 1)) # hide
```

--------------------------------

### Plot Density for Normal Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Normal distribution. Requires specifying the range and mean/standard deviation.

```julia
plotdensity((-4, 4), Normal, (0, 1)) # hide
```

--------------------------------

### Fit Normal Distribution to Data

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/starting.md

Estimate the parameters (mean and standard deviation) of a normal distribution that best fits a given set of data. This is useful for empirical modeling.

```julia
fit(Normal, x)
```

--------------------------------

### Plot Density for Chi Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Chi distribution. Requires specifying the range and degrees of freedom.

```julia
plotdensity((0.001, 3), Chi, (1,)) # hide
```

--------------------------------

### Plot Density for Exponential Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Exponential distribution. Requires specifying the range and rate parameter.

```julia
plotdensity((0, 3.5), Exponential, (0.5,)) # hide
```

--------------------------------

### Compute Sufficient Statistics

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/fit.md

Compute the sufficient statistics for a distribution `D` from data `x`. Supports weighted data.

```julia
ss = suffstats(D, x)
```

```julia
ss = suffstats(D, x, w)
```

--------------------------------

### Plot Density for Kumaraswamy Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Kumaraswamy distribution. Requires specifying the range and shape parameters.

```julia
plotdensity((0, 1), Kumaraswamy, (2, 5)) # hide
```

--------------------------------

### Plot Density for Rician Distribution

Source: https://github.com/juliastats/distributions.jl/blob/master/docs/src/univariate.md

Plots the density of the Rician distribution. Requires specifying the range and parameters.

```julia
plotdensity((0, 5), Rician, (0.5, 1)) # hide
```