---
sidebar_position: 3
title: Distribution
---

Distributions are the flagship data type in Squiggle. The distribution type is a generic data type that contains one of three different formats of distributions. 
These subtypes are [point set](/docs/Api/DistPointSet), [sample set](/docs/Api/DistSampleSet), and symbolic. The first two of these have a few custom functions that only work on them. You can read more about the differences between these formats [here](/docs/Discussions/Three-Formats-Of-Distributions). 

Several functions below only can work on particular distribution formats. 
For example, scoring and pointwise math requires the point set format. When this happens, the types are automatically converted to the correct format. These conversions are lossy.

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## Distribution Creation

These are functions for creating primative distributions. Many of these could optionally take in distributions as inputs; in these cases, Monte Carlo Sampling will be used to generate the greater distribution. This can be used for simple hierarchical models.

See a longer tutorial on creating distributions [here](/docs/Guides/DistributionCreation).

### normal

```
normal: (distribution|number, distribution|number) => distribution
normal: (dict<{p5: distribution|number, p95: distribution|number}>) => distribution
normal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution
```

**Examples**

```js
normal(5, 1)
normal({ p5: 4, p95: 10 })
normal({ mean: 5, stdev: 2 })
normal(5 to 10, normal(3, 2))
normal({ mean: uniform(5, 9), stdev: 3 })
```

### lognormal

```
lognormal: (distribution|number, distribution|number) => distribution
lognormal: (dict<{p5: distribution|number, p95: distribution|number}>) => distribution
lognormal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution
```

**Examples**

```javascript
lognormal(0.5, 0.8)
lognormal({ p5: 4, p95: 10 })
lognormal({ mean: 5, stdev: 2 })
```

### uniform

```
uniform: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
uniform(10, 12)
```

### beta

```
beta: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
beta(20, 25)
```

### cauchy

```
cauchy: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
cauchy(5, 1)
```

### gamma

```javascript
gamma: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
gamma(5, 1)
```

### Logistic

```
logistic: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
gamma(5, 1)
```

### exponential

```
exponential: (distribution|number) => distribution
```

**Examples**

```javascript
exponential(2)
```

### bernoulli

```
bernoulli: (distribution|number) => distribution
```

**Examples**

```javascript
bernoulli(0.5)
```

### triangular

```javascript
triangular: (number, number, number) => distribution
```

**Examples**

```javascript
triangular(5, 10, 20)
```

### To / credibleIntervalToDistribution

The `to` function is an easy way to generate simple distributions using predicted _5th_ and _95th_ percentiles.

If both values are above zero, a `lognormal` distribution is used. If not, a `normal` distribution is used.

``To`` is an alias for ``credibleIntervalToDistribution``. However, because of its frequent use, it is recommended to use the shorter name.

```
to: (distribution|number, distribution|number) => distribution
credibleIntervalToDistribution(distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
5 to 10
to(5,10)
-5 to 5
```


### mixture

```
mixture: (...distributionLike, weights?:list<float>) => distribution
mixture: (list<distributionLike>, weights?:list<float>) => distribution
```

**Examples**

```javascript
mixture(normal(5, 1), normal(10, 1), 8)
mx(normal(5, 1), normal(10, 1), [0.3, 0.7])
mx([normal(5, 1), normal(10, 1)], [0.3, 0.7])
```

## Functions

### sample

One random sample from the distribution

```
sample: (distribution) => number
```

**Examples**

```javascript
sample(normal(5, 2))
```

### sampleN

N random samples from the distribution

```
sampleN: (distribution, number) => list<number>
```

**Examples**

```javascript
sampleN(normal(5, 2), 100)
```

### mean

The distribution mean

```
mean: (distribution) => number
```

**Examples**

```javascript
mean(normal(5, 2))
```

### stdev

Standard deviation. Only works now on sample set distributions (so converts other distributions into sample set in order to calculate.)

```
stdev: (distribution) => number
```

### variance
Variance. Similar to stdev, only works now on sample set distributions.


```
variance: (distribution) => number
```

### mode

```
mode: (distribution) => number
```

### cdf

```
cdf: (distribution, number) => number
```

**Examples**

```javascript
cdf(normal(5, 2), 3)
```

### pdf

```
pdf: (distribution, number) => number
```

**Examples**

```javascript
pdf(normal(5, 2), 3)
```

### quantile

```
quantile: (distribution, number) => number
```

**Examples**

```javascript
quantile(normal(5, 2), 0.5)
```

### toPointSet

**TODO: Will soon be called "PointSet.make"**

Converts a distribution to the pointSet format

```
toPointSet: (distribution) => pointSetDistribution
```

**Examples**

```javascript
toPointSet(normal(5, 2))
```

### toSampleSet
**TODO: Will soon be called "SampleSet.make"**

Converts a distribution to the sampleSet format, with n samples

```
toSampleSet: (distribution, number) => sampleSetDistribution
```

**Examples**

```javascript
toSampleSet(normal(5, 2), 1000)
```

### truncateLeft

Truncates the left side of a distribution. Returns either a pointSet distribution or a symbolic distribution.

```
truncateLeft: (distribution, l => number) => distribution
```

**Examples**

```javascript
truncateLeft(normal(5, 2), 3)
```

### truncateRight

Truncates the right side of a distribution. Returns either a pointSet distribution or a symbolic distribution.

```
truncateRight: (distribution, r => number) => distribution
```

**Examples**

```javascript
truncateLeft(normal(5, 2), 6)
```

### klDivergence

[Kullback–Leibler divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) between two distributions.

```
klDivergence: (distribution, distribution) => number
```

**Examples**

```javascript
klDivergence(normal(5, 2), normal(5, 4)) // returns 0.57
```

## Display

### toString

```
toString: (distribution) => string
```

**Examples**

```javascript
toString(normal(5, 2))
```

### toSparkline

Produce a sparkline of length n

```
toSparkline: (distribution, n = 20) => string
```

**Examples**

```javascript
toSparkline(normal(5, 2), 10)
```

### inspect

Prints the value of the distribution to the Javascript console, then returns the distribution.

```
inspect: (distribution) => distribution
```

**Examples**

```javascript
inspect(normal(5, 2))
```

## Normalization

### normalize

Normalize a distribution. This means scaling it appropriately so that it's cumulative sum is equal to 1.

```
normalize: (distribution) => distribution
```

**Examples**

```javascript
normalize(normal(5, 2))
```

### isNormalized

Check of a distribution is normalized. Most distributions are typically normalized, but there are some commands that could produce non-normalized distributions.

```
isNormalized: (distribution) => bool
```

**Examples**

```javascript
isNormalized(normal(5, 2)) // returns true
```

### integralSum

Get the sum of the integral of a distribution. If the distribution is normalized, this will be 1.

```
integralSum: (distribution) => number
```

**Examples**

```javascript
integralSum(normal(5, 2))
```

## Regular Arithmetic Operations

Regular arithmetic operations cover the basic mathematical operations on distributions. They work much like their equivalent operations on numbers.

The infixes ``+``,``-``, ``*``, ``/``, ``^``, ``-`` are supported for addition, subtraction, multiplication, division, power, and unaryMinus.

```javascript
pointMass(5 + 10) == pointMass(5) + pointMass(10)
```

### add

```
add: (distributionLike, distributionLike) => distribution
```

**Examples**

```javascript
normal(0,1) + normal(1,3) // returns normal(1, 3.16...)
add(normal(0,1), normal(1,3)) // returns normal(1, 3.16...)
```

### sum
**Todo: Not yet implemented for distributions**

```
sum: (list<distributionLike>) => distribution
```

**Examples**

```javascript
sum([normal(0,1), normal(1,3), uniform(10,1)])
```

### multiply

```
multiply: (distributionLike, distributionLike) => distribution
```

### product

```
product: (list<distributionLike>) => distribution
```

### subtract

```
subtract: (distributionLike, distributionLike) => distribution
```

### divide

```
divide: (distributionLike, distributionLike) => distribution
```

### pow

```
pow: (distributionLike, distributionLike) => distribution
```

### exp

```
exp: (distributionLike, distributionLike) => distribution
```

### log

```
log: (distributionLike, distributionLike) => distribution
```

### log10

```
log10: (distributionLike, distributionLike) => distribution
```

### unaryMinus

```
unaryMinus: (distribution) => distribution
```

**Examples**

```javascript
-(normal(5,2)) // same as normal(-5, 2)
unaryMinus(normal(5,2)) // same as normal(-5, 2)
```

## Pointwise Arithmetic Operations

### dotAdd

```
dotAdd: (distributionLike, distributionLike) => distribution
```

### dotMultiply

```
dotMultiply: (distributionLike, distributionLike) => distribution
```

### dotSubtract

```
dotSubtract: (distributionLike, distributionLike) => distribution
```

### dotDivide

```
dotDivide: (distributionLike, distributionLike) => distribution
```

### dotPow

```
dotPow: (distributionLike, distributionLike) => distribution
```

### dotExp

```
dotExp: (distributionLike, distributionLike) => distribution
```

## Scale Arithmetic Operations

### scaleMultiply

```
scaleMultiply: (distributionLike, number) => distribution
```

### scalePow

```
scalePow: (distributionLike, number) => distribution
```

### scaleExp

```
scaleExp: (distributionLike, number) => distribution
```

### scaleLog

```
scaleLog: (distributionLike, number) => distribution
```

### scaleLog10

```
scaleLog10: (distributionLike, number) => distribution
```

## Special

### Declaration (Continuous Functions)

Adds metadata to a function of the input ranges. Works now for numeric and date inputs. This is useful when making predictions. It allows you to limit the domain that your prediction will be used and scored within.

```
declareFn: (dict<{fn: lambda, inputs: array<dict<{min: number, max: number}>>}>) => declaration
```

**Examples**

```javascript
declareFn({
  fn: {|a,b| a },
  inputs: [
    {min: 0, max: 100},
    {min: 30, max: 50}
  ]
})
```
