# Expectations

Installation (for Julia v1.0 and up):

```
pkg> add Expectations
```

See Pkg docs for more details

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
```