# Distribution Functions

## Standard Operations

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. The subtraction 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 multiplication 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 division 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.

### The base `e`

exponential

### Logarithms

A projection over a stretched x-axis.

Base `x`

## Pointwise Operations

### 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**.

### Pointwise multiplication

## Standard Functions

### 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 `quantile`

.

#### 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 statistics`

panel which appears
beneath distributions, the numbers beneath 5%, 10%, 25% etc are the quantiles of that distribution
for those precentage values.

#### 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

We can convert any distribution into the `SampleSet`

format

Or the `PointSet`

format

#### 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

#### 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