# Functions Reference

## 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.

### 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.

### 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.

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

### 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.

### 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.

### Taking the base `e`

exponential

### Taking logarithms

A projection over a stretched x-axis.

Base `x`

#### Validity

`x`

must be a scalar- See the current discourse

### 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.

### Pointwise subtraction

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

### Pointwise multiplication

### Pointwise division

### Pointwise exponentiation

## Standard functions on distributions

### Probability density function

The `pdf(dist, x)`

function returns the density of a distribution at the
given point x.

#### 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 `inv`

.

#### Validity

`x`

must be a scalar`dist`

must be a distribution

### Inverse CDF

The `inv(dist, prob)`

gives the value x or which 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.

#### 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.

### Sampling a distribution

The `sample(distribution)`

samples a given distribution.

## Converting between distribution formats

Recall the three formats of distributions. We can force any distribution into `SampleSet`

format

Or `PointSet`

format

`toSampleSet`

has two signatures

Above, we saw the unary `toSampleSet`

, which uses an internal hardcoded number of samples. If you'd like to provide the number of samples, it has a binary signature as well (floored)

#### Validity

- Second argument to
`toSampleSet`

must be a number.

## Normalization

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

We provide a `normalize`

function

#### Validity - Input to `normalize`

must be a dist

We provide a predicate `isNormalized`

, for when we have simple control flow

#### 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.

Save for a logging side effect, `inspect`

does nothing to input and returns it.

## Truncate

You can cut off from the left

You can cut off from the right

You can cut off from both sides