# Functions Reference

## Operating on distributions​

Here are the ways we combine distributions.

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 logarithms​

A projection over a stretched x-axis.

Base x

#### Validity​

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.

## 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