Getting Started

# Getting Started

## Installation

The Distributions package is available through the Julia package system by running `Pkg.add("Distributions")`. Throughout, we assume that you have installed the package.

## Starting With a Normal Distribution

We start by drawing 100 observations from a standard-normal random variable.

The first step is to set up the environment:

``````julia> using Random, Distributions

julia> Random.seed!(123) # Setting the seed``````

Then, we create a standard-normal distribution `d` and obtain samples using `rand`:

``````julia> d = Normal()
Normal(μ=0.0, σ=1.0)

julia> x = rand(d, 100)
100-element Array{Float64,1}:
0.376264
-0.405272
...``````

You can easily obtain the `pdf`, `cdf`, `quantile`, and many other functions for a distribution. For instance, the median (50th percentile) and the 95th percentile for the standard-normal distribution are given by:

``````julia> quantile.(Normal(), [0.5, 0.95])
2-element Array{Float64,1}:
0.0
1.64485``````

The normal distribution is parameterized by its mean and standard deviation. To draw random samples from a normal distribution with mean 1 and standard deviation 2, you write:

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

## Using Other Distributions

The package contains a large number of additional distributions of three main types:

• `Univariate`
• `Multivariate`
• `Matrixvariate`

Each type splits further into `Discrete` and `Continuous`.

For instance, you can define the following distributions (among many others):

``````julia> Binomial(p) # Discrete univariate
julia> Cauchy(u, b)  # Continuous univariate
julia> Multinomial(n, p) # Discrete multivariate
julia> Wishart(nu, S) # Continuous matrix-variate``````

In addition, you can create truncated distributions from univariate distributions:

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

To find out which parameters are appropriate for a given distribution `D`, you can use `fieldnames(D)`:

``````julia> fieldnames(Cauchy)
2-element Array{Symbol,1}:
:μ
:β``````

This tells you that a Cauchy distribution is initialized with location `μ` and scale `β`.

## Estimate the Parameters

It is often useful to approximate an empirical distribution with a theoretical distribution. As an example, we can use the array `x` we created above and ask which normal distribution best describes it:

``````julia> fit(Normal, x)
Normal(μ=0.036692077201688635, σ=1.1228280164716382)``````

Since `x` is a random draw from `Normal`, it's easy to check that the fitted values are sensible. Indeed, the estimates [0.04, 1.12] are close to the true values of [0.0, 1.0] that we used to generate `x`.