# 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)

The object d represents a probability distribution, in our case the standard-normal distribution. One can query its properties such as the mean:

julia> mean(d)
0.0

We can also draw samples from d with rand.

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 == ArrayLikeVariate{0}
• Multivariate == ArrayLikeVariate{1}
• Matrixvariate == ArrayLikeVariate{2}

Each type splits further into Discrete and Continuous.

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

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

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.