---
title: "Distributions: Key Functions"
sidebar_position: 3
---

import { SquiggleEditor } from "../../src/components/SquiggleEditor";

## Operating on distributions

Here are the ways we combine distributions.

### Addition

A horizontal right shift. The addition operation represents the distribution of the sum of
the value of one random sample chosen from the first distribution and the value one random sample
chosen from the second distribution.

<SquiggleEditor
  defaultCode={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 + dist2`}
/>

### Subtraction

A horizontal left shift. A horizontal right shift. The substraction operation represents
the distribution of the value of one random sample chosen from the first distribution minus
the value of one random sample chosen from the second distribution.

<SquiggleEditor
  defaultCode={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 - dist2`}
/>

### Multiplication

A proportional scaling. The addition operation represents the distribution of the multiplication of
the value of one random sample chosen from the first distribution times the value one random sample
chosen from the second distribution.

<SquiggleEditor
  defaultCode={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 * dist2`}
/>

We also provide concatenation of two distributions as a syntax sugar for `*`

<SquiggleEditor defaultCode="(0.1 to 1) * triangular(1,2,3)" />

### Division

A proportional scaling (normally a shrinking if the second distribution has values higher than 1).
The addition operation represents the distribution of the division of
the value of one random sample chosen from the first distribution over the value one random sample
chosen from the second distribution. If the second distribution has some values near zero, it
tends to be particularly unstable.

<SquiggleEditor
  defaultCode={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 / dist2`}
/>

### Exponentiation

A projection over a contracted x-axis. The exponentiation operation represents the distribution of
the exponentiation of the value of one random sample chosen from the first distribution to the power of
the value one random sample chosen from the second distribution.

<SquiggleEditor defaultCode={`(0.1 to 1) ^ beta(2, 3)`} />

### Taking the base `e` exponential

<SquiggleEditor
  defaultCode={`dist = triangular(1,2,3)
exp(dist)`}
/>

### Taking logarithms

A projection over a stretched x-axis.

<SquiggleEditor
  defaultCode={`dist = triangular(1,2,3)
log(dist)`}
/>

<SquiggleEditor defaultCode={`log10(5 to 10)`} />

Base `x`

<SquiggleEditor defaultCode={`log(5 to 10, 2)`} />

#### Validity

- `x` must be a scalar
- See [the current discourse](https://github.com/quantified-uncertainty/squiggle/issues/304)

### Pointwise addition

For every point on the x-axis, operate the corresponding points in the y axis of the pdf.

**Pointwise operations are done with `PointSetDist` internals rather than `SampleSetDist` internals**.

TODO: this isn't in the new interpreter/parser yet.

<SquiggleEditor defaultCode={`(1 to 10) .+ triangular(1,2,3)`} />

### Pointwise subtraction

TODO: this isn't in the new interpreter/parser yet.

<SquiggleEditor defaultCode={`(1 to 10) .- triangular(1,2,3)`} />

### Pointwise multiplication

<SquiggleEditor defaultCode={`(1 to 10) .* triangular(1,2,3)`} />

### Pointwise division

<SquiggleEditor defaultCode={`(uniform(0,10)) ./ normal(10,4)`} />

### Pointwise exponentiation

<SquiggleEditor defaultCode={`(1 to 10) .^ triangular(1,2,3)`} />

## Standard functions on distributions

### Probability density function

The `pdf(dist, x)` function returns the density of a distribution at the
given point x.

<SquiggleEditor defaultCode="pdf(normal(0,1),0)" />

#### Validity

- `x` must be a scalar
- `dist` must be a distribution

### Cumulative density function

The `cdf(dist, x)` gives the cumulative probability of the distribution
or all values lower than x. It is the inverse of `quantile`.

<SquiggleEditor defaultCode="cdf(normal(0,1),0)" />

#### Validity

- `x` must be a scalar
- `dist` must be a distribution

### Quantile

The `quantile(dist, prob)` gives the value x for which the sum of the probability for all values
lower than x is equal to prob. It is the inverse of `cdf`. In the literature, it
is also known as the quantiles function. In the optional `summary stistics` panel which appears
beneath distributions, the numbers beneath 5%, 10%, 25% etc are the quantiles of that distribution
for those precentage values.

<SquiggleEditor defaultCode="quantile(normal(0,1),0.5)" />

#### Validity

- `prob` must be a scalar (please only put it in `(0,1)`)
- `dist` must be a distribution

### Mean

The `mean(distribution)` function gives the mean (expected value) of a distribution.

<SquiggleEditor defaultCode="mean(normal(5, 10))" />

### Sampling a distribution

The `sample(distribution)` samples a given distribution.

<SquiggleEditor defaultCode="sample(normal(0, 10))" />

## Converting between distribution formats

Recall the [three formats of distributions](/docs/Discussions/Three-Formats-Of-Distributions). We can force any distribution into `SampleSet` format

<SquiggleEditor defaultCode="SampleSet.fromDist(normal(5, 10))" />

Or `PointSet` format

<SquiggleEditor defaultCode="PointSet.fromDist(normal(5, 10))" />

#### Validity

- Second argument to `SampleSet.fromDist` must be a number.

## Normalization

Some distribution operations (like horizontal shift) return an unnormalized distriibution.

We provide a `normalize` function

<SquiggleEditor defaultCode="normalize((0.1 to 1) + triangular(0.1, 1, 10))" />

#### Validity - Input to `normalize` must be a dist

We provide a predicate `isNormalized`, for when we have simple control flow

<SquiggleEditor defaultCode="isNormalized((0.1 to 1) * triangular(0.1, 1, 10))" />

#### Validity

- Input to `isNormalized` must be a dist

## `inspect`

You may like to debug by right clicking your browser and using the _inspect_ functionality on the webpage, and viewing the _console_ tab. Then, wrap your squiggle output with `inspect` to log an internal representation.

<SquiggleEditor defaultCode="inspect(SampleSet.fromDist(0.1 to 1))" />

Save for a logging side effect, `inspect` does nothing to input and returns it.

## Truncate

You can cut off from the left

<SquiggleEditor defaultCode="truncateLeft(0.1 to 1, 0.5)" />

You can cut off from the right

<SquiggleEditor defaultCode="truncateRight(0.1 to 1, 0.5)" />

You can cut off from both sides

<SquiggleEditor defaultCode="truncate(0.1 to 1, 0.5, 1.5)" />