## Expectations.jl

Expectation operators for Distributions.jl objects
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July 2018 # Expectations

Installation (for Julia v1.0 and up):

`pkg> add Expectations`

This is a package designed to simplify the process of taking expectations of functions of random variables.

### Expectation Operator

The key object is the `expectation` function, which returns an operator:

```dist = Normal()
E = expectation(dist)
E(x -> x)```

For convenience, the operator can be applied directly to a function instead of being cached,

`expectation(x->x^2, dist)`

As a linear operator on vectors using the nodes of the distribution

```dist = Normal()
E = expectation(dist)
x = nodes(E)
f(x) = x^2
E * f.(x) == dot(f.(x), weights(E))```

### Random Variables

The underlying distributions are objects from `Distributions.jl` (currently `<:UnivariateDistribution`).

Starting with 1.3.0, we also support mixture models.

### Quadrature Algorithms

We support different types of Gaussian quadrature (Gauss-Hermite, Gauss-Legendre, Gauss-Laguerre, etc.) based on the distribution, as well as some methods with user-defined nodes (e.g., trapezoidal integration).

We have rules for the following distributions:

• Normal
• ChiSq
• LogNormal
• Exponential
• Beta
• Gamma/Erlang
• Uniform
• Continuous Univariate (compact; generic fallback)
• Continuous Univariate (no restriction; approximates with quantile grid)
• Discrete

See docs for more info.

### Mixture Models

We also support mixture models, e.g.

```d = MixtureModel([Uniform(), Normal(), Gamma()]);
E = expectation(d);
E(x -> x) # 0.5000000000000016```

The `MixtureExpectation` objects support most of the same behavior as the individual `IterableExpectation`s.

```2E(x -> x) # 1.000000000000003
weights(E) # [1/3, 1/3, 1/3]
expectations(E) # component expectations```