# Distribution Creation

## Normal

```
normal(mean: number, standardDeviation: number)
normal({mean: number, standardDeviation: number})
normal({p5: number, p95: number})
normal({p10: number, p90: number})
normal({p25: number, p75: number})
```

Creates a normal distribution (opens in a new tab) with the given mean and standard deviation.

Wikipedia (opens in a new tab)

## Lognormal

```
lognormal(mu: number, sigma: number)
lognormal({mean: number, standardDeviation: number})
lognormal({p5: number, p95: number})
lognormal({p10: number, p90: number})
lognormal({p25: number, p75: number})
```

Creates a lognormal distribution (opens in a new tab) with the given mu and sigma.

`Mu`

and `sigma`

represent the mean and standard deviation of the normal which results when
you take the log of our lognormal distribution. They can be difficult to directly reason about. However, there are several alternative ways to specify a lognormal distribution which are often easier to reason about.

Wikipedia (opens in a new tab)

❓ Understanding mu and sigma

The log of `lognormal(mu, sigma)`

is a normal distribution with
mean `mu`

and standard deviation `sigma`

. For example, these two distributions
are identical:

## To

```
(5thPercentile: number) to (95thPercentile: number)
to(5thPercentile: number, 95thPercentile: number)
```

The `to`

function is an easy way to generate lognormal distributions using predicted *5th* and *95th* percentiles. It's the same as `lognormal({p5, p95})`

, but easier to write and read.

### Arguments

`5thPercentile`

: number`95thPercentile`

: number, greater than`5thPercentile`

**Tip**

"

**Caution**

If you haven't tried calibration
training (opens in a new tab),
you're likely to be overconfident. We recommend doing calibration training
to get a feel for what a 90 percent confident interval feels like.

## Uniform

`uniform(low:number, high:number)`

Creates a uniform distribution (opens in a new tab) with the given low and high values.

### Arguments

`low`

: Number`high`

: Number greater than`low`

**Caution**

While uniform distributions are very simple to understand, we find it rare
to find uncertainties that actually look like this. Before using a uniform
distribution, think hard about if you are really 100% confident that the
paramater will not wind up being just outside the stated boundaries.

One good example of a uniform distribution uncertainty would be clear
physical limitations. You might have complete complete uncertainty on what
time of day an event will occur, but can say with 100% confidence it will
happen between the hours of 0:00 and 24:00.

## Point Mass

`pointMass(value:number)`

Creates a discrete distribution with all of its probability mass at point `value`

.

Few Squiggle users call the function `pointMass()`

directly. Numbers are often (but not always) converted into point mass distributions automatically, when it is appropriate.

For example, in the function `mixture(1,2,normal(5,2))`

, the first two arguments will get converted into point mass distributions
with values at 1 and 2. Therefore, this is the same as `mixture(pointMass(1),pointMass(2),pointMass(5,2))`

.

`pointMass()`

distributions are currently the only discrete distributions accessible in Squiggle.

### Arguments

`value`

: Number

## Beta

```
beta(alpha:number, beta:number)
beta({mean: number, stdev: number})
```

Creates a beta distribution (opens in a new tab) with the given `alpha`

and `beta`

values. For a good summary of the beta distribution, see this explanation (opens in a new tab) on Stack Overflow.

### Arguments

`alpha`

: Number greater than zero`beta`

: Number greater than zero

**Caution with small numbers**

Squiggle struggles to show beta distributions when either alpha or beta are
below 1.0. This is because the tails at ~0.0 and ~1.0 are very high. Using a
log scale for the y-axis helps here.

## Examples

## Mixture

```
mixture(...distributions: Distribution[], weights?: number[])
mixture(distributions: Distribution[], weights?: number[])
mx(...distributions: Distribution[], weights?: number[])
mx(distributions: Distribution[], weights?: number[])
```

The `mixture`

mixes combines multiple distributions to create a mixture. You can optionally pass in a list of proportional weights.

### Arguments

`distributions`

: A set of distributions or numbers, each passed as a paramater. Numbers will be converted into point mass distributions.`weights`

: An optional array of numbers, each representing the weight of its corresponding distribution. The weights will be re-scaled to add to`1.0`

. If a weights array is provided, it must be the same length as the distribution paramaters.

### Aliases

`mx`

### Special Use Cases of Mixtures

## 🕐 Zero or Continuous

One common reason to have mixtures of continous and discrete distributions is to handle the special case of 0. Say I want to model the time I will spend on some upcoming project. I think I have an 80% chance of doing it.

In this case, I have a 20% chance of spending 0 time with it. I might estimate my hours with,

## 🔒 Model Uncertainty Safeguarding

One technique several Foretold.io users used is to combine their main guess, with a "just-in-case distribution". This latter distribution would have very low weight, but would be very wide, just in case they were dramatically off for some weird reason.

## SampleSet.fromList

`SampleSet.fromList(samples:number[])`

Creates a sample set distribution using an array of samples.

Samples are converted into PDFs automatically using kernel density estimation (opens in a new tab) and an approximated bandwidth. This is an approximation and can be error-prone.

### Arguments

`samples`

: An array of at least 5 numbers.

## PointSet.makeContinuous

`PointSet.makeContinuous(points:{x: number, y: number})`

Creates a continuous point set distribution using a list of points.

**Caution!**

Distributions made with `makeContinuous`

are not automatically normalized. We suggest normalizing them manually using the `normalize`

function.

### Arguments

`points`

: An array of at least 3 coordinates.

## PointSet.makeDiscrete

`PointSet.makeDiscrete(points:{x: number, y: number})`

Creates a discrete point set distribution using a list of points.

### Arguments

`points`

: An array of at least 1 coordinate.