369 lines
8.9 KiB
Plaintext
369 lines
8.9 KiB
Plaintext
---
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title: "Functions Reference"
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sidebar_position: 7
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---
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import { SquiggleEditor } from "../../src/components/SquiggleEditor";
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_The source of truth for this document is [this file of code](https://github.com/quantified-uncertainty/squiggle/blob/develop/packages/squiggle-lang/src/rescript/ReducerInterface/ReducerInterface_GenericDistribution.res)_
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## Inventory distributions
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We provide starter distributions, computed symbolically.
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### Normal distribution
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The `normal(mean, sd)` function creates a normal distribution with the given mean
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and standard deviation.
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<SquiggleEditor initialSquiggleString="normal(5, 1)" />
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#### Validity
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- `sd > 0`
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### Uniform distribution
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The `uniform(low, high)` function creates a uniform distribution between the
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two given numbers.
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<SquiggleEditor initialSquiggleString="uniform(3, 7)" />
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#### Validity
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- `low < high`
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### Lognormal distribution
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The `lognormal(mu, sigma)` returns the log of a normal distribution with parameters
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`mu` and `sigma`. The log of `lognormal(mu, sigma)` is a normal distribution with mean `mu` and standard deviation `sigma`.
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<SquiggleEditor initialSquiggleString="lognormal(0, 0.7)" />
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An alternative format is also available. The `to` notation creates a lognormal
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distribution with a 90% confidence interval between the two numbers. We add
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this convenience as lognormal distributions are commonly used in practice.
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<SquiggleEditor initialSquiggleString="2 to 10" />
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#### Future feature:
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Furthermore, it's also possible to create a lognormal from it's actual mean
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and standard deviation, using `lognormalFromMeanAndStdDev`.
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TODO: interpreter/parser doesn't provide this in current `develop` branch
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<SquiggleEditor initialSquiggleString="lognormalFromMeanAndStdDev(20, 10)" />
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#### Validity
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- `sigma > 0`
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- In `x to y` notation, `x < y`
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### Beta distribution
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The `beta(a, b)` function creates a beta distribution with parameters `a` and `b`:
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<SquiggleEditor initialSquiggleString="beta(10, 20)" />
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#### Validity
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- `a > 0`
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- `b > 0`
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- Empirically, we have noticed that numerical instability arises when `a < 1` or `b < 1`
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### Exponential distribution
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The `exponential(rate)` function creates an exponential distribution with the given
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rate.
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<SquiggleEditor initialSquiggleString="exponential(1.11)" />
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#### Validity
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- `rate > 0`
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### Triangular distribution
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The `triangular(a,b,c)` function creates a triangular distribution with lower
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bound `a`, mode `b` and upper bound `c`.
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#### Validity
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- `a < b < c`
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<SquiggleEditor initialSquiggleString="triangular(1, 2, 4)" />
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### Scalar (constant dist)
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Squiggle, when the context is right, automatically casts a float to a constant distribution.
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## Operating on distributions
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Here are the ways we combine distributions.
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### Mixture of distributions
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The `mixture` function combines 2 or more other distributions to create a weighted
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combination of the two. The first positional arguments represent the distributions
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to be combined, and the last argument is how much to weigh every distribution in the
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combination.
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<SquiggleEditor initialSquiggleString="mixture(uniform(0,1), normal(1,1), [0.5, 0.5])" />
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It's possible to create discrete distributions using this method.
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<SquiggleEditor initialSquiggleString="mixture(0, 1, [0.2,0.8])" />
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As well as mixed distributions:
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<SquiggleEditor initialSquiggleString="mixture(3, 8, 1 to 10, [0.2, 0.3, 0.5])" />
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An alias of `mixture` is `mx`
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#### Validity
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Using javascript's variable arguments notation, consider `mx(...dists, weights)`:
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- `dists.length == weights.length`
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### Addition
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A horizontal right shift
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 + dist2`}
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/>
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### Subtraction
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A horizontal left shift
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 - dist2`}
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/>
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### Multiplication
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TODO: provide intuition pump for the semantics
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 * dist2`}
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/>
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We also provide concatenation of two distributions as a syntax sugar for `*`
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<SquiggleEditor initialSquiggleString="(0.1 to 1) triangular(1,2,3)" />
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### Division
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TODO: provide intuition pump for the semantics
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 / dist2`}
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/>
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### Exponentiation
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TODO: provide intuition pump for the semantics
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<SquiggleEditor initialSquiggleString={`(0.1 to 1) ^ beta(2, 3)`} />
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### Taking the base `e` exponential
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<SquiggleEditor
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initialSquiggleString={`dist = triangular(1,2,3)
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exp(dist)`}
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/>
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### Taking logarithms
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<SquiggleEditor
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initialSquiggleString={`dist = triangular(1,2,3)
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log(dist)`}
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/>
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<SquiggleEditor
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initialSquiggleString={`dist = beta(1,2)
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log10(dist)`}
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/>
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Base `x`
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<SquiggleEditor
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initialSquiggleString={`x = 2
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dist = beta(2,3)
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log(dist, x)`}
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/>
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#### Validity
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- `x` must be a scalar
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- See [the current discourse](https://github.com/quantified-uncertainty/squiggle/issues/304)
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### Pointwise addition
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**Pointwise operations are done with `PointSetDist` internals rather than `SampleSetDist` internals**.
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TODO: this isn't in the new interpreter/parser yet.
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .+ dist2`}
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/>
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### Pointwise subtraction
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TODO: this isn't in the new interpreter/parser yet.
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .- dist2`}
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/>
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### Pointwise multiplication
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .* dist2`}
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/>
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### Pointwise division
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 ./ dist2`}
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/>
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### Pointwise exponentiation
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<SquiggleEditor
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initialSquiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .^ dist2`}
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/>
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## Standard functions on distributions
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### Probability density function
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The `pdf(dist, x)` function returns the density of a distribution at the
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given point x.
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<SquiggleEditor initialSquiggleString="pdf(normal(0,1),0)" />
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#### Validity
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- `x` must be a scalar
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- `dist` must be a distribution
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### Cumulative density function
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The `cdf(dist, x)` gives the cumulative probability of the distribution
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or all values lower than x. It is the inverse of `inv`.
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<SquiggleEditor initialSquiggleString="cdf(normal(0,1),0)" />
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#### Validity
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- `x` must be a scalar
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- `dist` must be a distribution
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### Inverse CDF
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The `inv(dist, prob)` gives the value x or which the probability for all values
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lower than x is equal to prob. It is the inverse of `cdf`.
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<SquiggleEditor initialSquiggleString="inv(normal(0,1),0.5)" />
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#### Validity
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- `prob` must be a scalar (please only put it in `(0,1)`)
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- `dist` must be a distribution
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### Mean
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The `mean(distribution)` function gives the mean (expected value) of a distribution.
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<SquiggleEditor initialSquiggleString="mean(normal(5, 10))" />
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### Sampling a distribution
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The `sample(distribution)` samples a given distribution.
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<SquiggleEditor initialSquiggleString="sample(normal(0, 10))" />
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## Converting between distribution formats
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Recall the [three formats of distributions](https://develop--squiggle-documentation.netlify.app/docs/Discussions/Three-Types-Of-Distributions). We can force any distribution into `SampleSet` format
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<SquiggleEditor initialSquiggleString="toSampleSet(normal(5, 10))" />
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Or `PointSet` format
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<SquiggleEditor initialSquiggleString="toPointSet(normal(5, 10))" />
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## Normalization
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Some distribution operations (like horizontal shift) return an unnormalized distriibution.
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We provide a `normalize` function
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<SquiggleEditor initialSquiggleString="normalize((0.1 to 1) + triangular(0.1, 1, 10))" />
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#### Validity - Input to `normalize` must be a dist
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We provide a predicate `isNormalized`, for when we have simple control flow
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<SquiggleEditor initialSquiggleString="isNormalized((0.1 to 1) * triangular(0.1, 1, 10))" />
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#### Validity
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- Input to `isNormalized` must be a dist
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## Convert any distribution to a sample set distribution
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`toSampleSet` has two signatures
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It is unary when you use an internal hardcoded number of samples
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<SquiggleEditor initialSquiggleString="toSampleSet(0.1 to 1)" />
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And binary when you provide a number of samples (floored)
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<SquiggleEditor initialSquiggleString="toSampleSet(0.1 to 1, 100)" />
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## `inspect`
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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.
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<SquiggleEditor initialSquiggleString="inspect(toSampleSet(0.1 to 1, 100))" />
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Save for a logging side effect, `inspect` does nothing to input and returns it.
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## Truncate
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You can cut off from the left
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<SquiggleEditor initialSquiggleString="truncateLeft(0.1 to 1, 0.5)" />
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You can cut off from the right
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<SquiggleEditor initialSquiggleString="truncateRight(0.1 to 1, 10)" />
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You can cut off from both sides
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<SquiggleEditor initialSquiggleString="truncate(0.1 to 1, 0.5, 1.5)" />
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