411 lines
13 KiB
Plaintext
411 lines
13 KiB
Plaintext
---
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title: "Distributions: Creation"
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sidebar_position: 2
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---
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import { SquiggleEditor } from "../../src/components/SquiggleEditor";
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import Admonition from "@theme/Admonition";
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import Tabs from "@theme/Tabs";
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import TabItem from "@theme/TabItem";
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## To
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`(5thPercentile: number) to (95thPercentile: number)`
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`to(5thPercentile: number, 95thPercentile: number)`
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The `to` function is an easy way to generate simple distributions using predicted _5th_ and _95th_ percentiles.
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If both values are above zero, a `lognormal` distribution is used. If not, a `normal` distribution is used.
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<Tabs lazy>
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<TabItem value="ex1" label="5 to 10" default>
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When <code>5 to 10</code> is entered, both numbers are positive, so it
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generates a lognormal distribution with 5th and 95th percentiles at 5 and
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10.
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<SquiggleEditor defaultCode="5 to 10" />
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</TabItem>
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<TabItem value="ex3" label="to(5,10)">
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<code>5 to 10</code> does the same thing as <code>to(5,10)</code>.
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<SquiggleEditor defaultCode="to(5,10)" />
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</TabItem>
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<TabItem value="ex2" label="-5 to 5">
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When <code>-5 to 5</code> is entered, there's negative values, so it
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generates a normal distribution. This has 5th and 95th percentiles at 5 and
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10.
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<SquiggleEditor defaultCode="-5 to -3" />
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</TabItem>
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<TabItem value="ex4" label="1 to 10000">
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It's very easy to generate distributions with very long tails. These can be
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impossible to see without changing view settings. (These settings are
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available in the Playground, but not this smaller editor component)
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<SquiggleEditor defaultCode="1 to 10000" />
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</TabItem>
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</Tabs>
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### Arguments
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- `5thPercentile`: number
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- `95thPercentile`: number, greater than `5thPercentile`
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<Admonition type="tip" title="Tip">
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<p>
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"<bold>To</bold>" is a great way to generate probability distributions very
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quickly from your intuitions. It's easy to write and easy to read. It's
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often a good place to begin an estimate.
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</p>
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</Admonition>
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<Admonition type="caution" title="Caution">
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<p>
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If you haven't tried{" "}
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<a href="https://www.lesswrong.com/posts/LdFbx9oqtKAAwtKF3/list-of-probability-calibration-exercises">
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calibration training
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</a>
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, you're likely to be overconfident. We recommend doing calibration training
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to get a feel for what a 90 percent confident interval feels like.
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</p>
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</Admonition>
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## Mixture
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`mixture(...distributions: Distribution[], weights?: number[])`
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`mx(...distributions: Distribution[], weights?: number[])`
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`mixture(distributions: Distributions[], weights?: number[])`
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`mx(distributions: Distributions[], weights?: number[])`
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The `mixture` mixes combines multiple distributions to create a mixture. You can optionally pass in a list of proportional weights.
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<Tabs lazy>
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<TabItem value="ex1" label="Simple" default>
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<SquiggleEditor defaultCode="mixture(1 to 2, 5 to 8, 9 to 10)" />
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</TabItem>
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<TabItem value="ex2" label="With Weights">
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<SquiggleEditor defaultCode="mixture(1 to 2, 5 to 8, 9 to 10, [0.1, 0.1, 0.8])" />
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</TabItem>
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<TabItem value="ex3" label="With Continuous and Discrete Inputs">
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<SquiggleEditor defaultCode="mixture(1 to 5, 8 to 10, 1, 3, 20)" />
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</TabItem>
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<TabItem value="ex4" label="Array of Distributions Input">
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<SquiggleEditor defaultCode="mx([1 to 2, exponential(1)], [1,1])" />
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</TabItem>
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</Tabs>
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### Arguments
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- `distributions`: A set of distributions or numbers, each passed as a paramater. Numbers will be converted into point mass distributions.
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- `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.
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### Aliases
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- `mx`
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### Special Use Cases of Mixtures
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<details>
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<summary>🕐 Zero or Continuous</summary>
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<p>
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One common reason to have mixtures of continous and discrete distributions is to handle the special case of 0.
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Say I want to model the time I will spend on some upcoming project. I think I have an 80% chance of doing it.
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</p>
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<p>
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In this case, I have a 20% chance of spending 0 time with it. I might estimate my hours with,
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</p>
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<Admonition type="caution" title="Caution">
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<p>
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There's a temporary bug where the below render is compressed. If you toggle the code it will fix render correctly.
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</p>
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</Admonition>
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<SquiggleEditor
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defaultCode={`hours_the_project_will_take = 5 to 20
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chance_of_doing_anything = 0.8
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mx(hours_the_project_will_take, 0, [chance_of_doing_anything, 1 - chance_of_doing_anything])`}
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/>
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</details>
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<details>
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<summary>🔒 Model Uncertainty Safeguarding</summary>
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<p>
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One technique several <a href="https://www.foretold.io/">Foretold.io</a> users used is to combine their main guess, with a
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"just-in-case distribution". This latter distribution would have very low weight, but would be
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very wide, just in case they were dramatically off for some weird reason.
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</p>
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<SquiggleEditor
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defaultCode={`forecast = 3 to 30
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chance_completely_wrong = 0.05
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forecast_if_completely_wrong = -100 to 200
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mx(forecast, forecast_if_completely_wrong, [1-chance_completely_wrong, chance_completely_wrong])`}
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/>
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</details>
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## Normal
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`normal(mean:number, standardDeviation:number)`
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Creates a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution) with the given mean and standard deviation.
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<Tabs lazy>
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<TabItem value="ex1" label="normal(5,1)" default>
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<SquiggleEditor defaultCode="normal(5, 1)" />
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</TabItem>
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<TabItem value="ex2" label="normal(1G, 1G)">
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<SquiggleEditor defaultCode="normal(1G, 1G)" />
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</TabItem>
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</Tabs>
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### Arguments
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- `mean`: Number
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- `standard deviation`: Number greater than zero
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[Wikipedia](https://en.wikipedia.org/wiki/Normal_distribution)
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## Log-normal
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`lognormal(mu: number, sigma: number)`
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Creates a [log-normal distribution](https://en.wikipedia.org/wiki/Log-normal_distribution) with the given mu and sigma.
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`Mu` and `sigma` represent the mean and standard deviation of the normal which results when
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you take the log of our lognormal distribution. They can be difficult to directly reason about.
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Because of this complexity, we recommend typically using the <a href="#to">to</a> syntax instead of estimating `mu` and `sigma` directly.
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<SquiggleEditor defaultCode="lognormal(0, 0.7)" />
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### Arguments
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- `mu`: Number
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- `sigma`: Number greater than zero
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[Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution)
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<details>
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<summary>
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❓ Understanding <bold>mu</bold> and <bold>sigma</bold>
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</summary>
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<p>
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The log of <code>lognormal(mu, sigma)</code> is a normal distribution with
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mean <code>mu</code>
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and standard deviation <code>sigma</code>. For example, these two distributions
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are identical:
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</p>
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<SquiggleEditor
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defaultCode={`normalMean = 10
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normalStdDev = 2
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logOfLognormal = log(lognormal(normalMean, normalStdDev))
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[logOfLognormal, normal(normalMean, normalStdDev)]`}
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/>
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</details>
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## Uniform
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`uniform(low:number, high:number)`
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Creates a [uniform distribution](<https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)>) with the given low and high values.
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<SquiggleEditor defaultCode="uniform(3,7)" />
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### Arguments
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- `low`: Number
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- `high`: Number greater than `low`
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<Admonition type="caution" title="Caution">
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<p>
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While uniform distributions are very simple to understand, we find it rare
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to find uncertainties that actually look like this. Before using a uniform
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distribution, think hard about if you are really 100% confident that the
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paramater will not wind up being just outside the stated boundaries.
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</p>
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<p>
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One good example of a uniform distribution uncertainty would be clear
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physical limitations. You might have complete complete uncertainty on what
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time of day an event will occur, but can say with 100% confidence it will
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happen between the hours of 0:00 and 24:00.
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</p>
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</Admonition>
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## Point Mass
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`pointMass(value:number)`
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Creates a discrete distribution with all of its probability mass at point `value`.
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Few Squiggle users call the function `pointMass()` directly. Numbers are converted into point mass distributions automatically, when it is appropriate.
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For example, in the function `mixture(1,2,normal(5,2))`, the first two arguments will get converted into point mass distributions
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with values at 1 and 2. Therefore, this is the same as `mixture(pointMass(1),pointMass(2),pointMass(5,2))`.
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`pointMass()` distributions are currently the only discrete distributions accessible in Squiggle.
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<Tabs lazy>
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<TabItem value="ex1" label="pointMass(3)" default>
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<SquiggleEditor defaultCode="pointMass(3)" />
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</TabItem>
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<TabItem value="ex3" label="mixture(1,3,5)">
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<SquiggleEditor defaultCode="mixture(1,3,5)" />
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</TabItem>
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<TabItem value="ex2" label="normal(5,2) * 6">
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<SquiggleEditor defaultCode="normal(5,2) * 6" />
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</TabItem>
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<TabItem value="ex4" label="dotAdd(normal(5,2), 6)">
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<SquiggleEditor defaultCode="dotAdd(normal(5,2), 6)" />
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</TabItem>
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<TabItem value="ex5" label="dotMultiply(normal(5,2), 6)">
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<SquiggleEditor defaultCode="dotMultiply(normal(5,2), 6)" />
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</TabItem>
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</Tabs>
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### Arguments
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- `value`: Number
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## Beta
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`beta(alpha:number, beta:number)`
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`beta({mean: number, stdev: number})`
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Creates a [beta distribution](https://en.wikipedia.org/wiki/Beta_distribution) with the given `alpha` and `beta` values. For a good summary of the beta distribution, see [this explanation](https://stats.stackexchange.com/a/47782) on Stack Overflow.
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<Tabs lazy>
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<TabItem value="ex1" label="beta(10, 20)" default>
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<SquiggleEditor defaultCode="beta(10,20)" />
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</TabItem>
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<TabItem value="ex2" label="beta(1000, 1000)">
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<SquiggleEditor defaultCode="beta(1000, 2000)" />
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</TabItem>
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<TabItem value="ex3" label="beta(1, 10)">
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<SquiggleEditor defaultCode="beta(1, 10)" />
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</TabItem>
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<TabItem value="ex4" label="beta(10, 1)">
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<SquiggleEditor defaultCode="beta(10, 1)" />
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</TabItem>
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<TabItem value="ex5" label="beta(0.8, 0.8)">
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<SquiggleEditor defaultCode="beta(0.8, 0.8)" />
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</TabItem>
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<TabItem value="ex6" label="beta({mean: 0.39, stdev: 0.1})">
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<SquiggleEditor defaultCode="beta({mean: 0.39, stdev: 0.1})" />
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</TabItem>
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</Tabs>
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### Arguments
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- `alpha`: Number greater than zero
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- `beta`: Number greater than zero
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<Admonition type="caution" title="Caution with small numbers">
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<p>
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Squiggle struggles to show beta distributions when either alpha or beta are
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below 1.0. This is because the tails at ~0.0 and ~1.0 are very high. Using a
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log scale for the y-axis helps here.
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</p>
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<details>
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<summary>Examples</summary>
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<Tabs lazy>
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<TabItem value="ex1" label="beta(0.3, 0.3)" default>
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<SquiggleEditor defaultCode="beta(0.3, 0.3)" />
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</TabItem>
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<TabItem value="ex2" label="beta(0.5, 0.5)">
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<SquiggleEditor defaultCode="beta(0.5, 0.5)" />
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</TabItem>
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<TabItem value="ex3" label="beta(0.8, 0.8)">
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<SquiggleEditor defaultCode="beta(.8,.8)" />
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</TabItem>
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<TabItem value="ex4" label="beta(0.9, 0.9)">
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<SquiggleEditor defaultCode="beta(.9,.9)" />
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</TabItem>
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</Tabs>
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</details>
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</Admonition>
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## Exponential
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`exponential(rate:number)`
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Creates an [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution) with the given rate.
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<SquiggleEditor defaultCode="exponential(4)" />
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### Arguments
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- `rate`: Number greater than zero
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## Triangular distribution
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`triangular(low:number, mode:number, high:number)`
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Creates a [triangular distribution](https://en.wikipedia.org/wiki/Triangular_distribution) with the given low, mode, and high values.
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### Arguments
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- `low`: Number
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- `mode`: Number greater than `low`
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- `high`: Number greater than `mode`
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<SquiggleEditor defaultCode="triangular(1, 2, 4)" />
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## SampleSet.fromList
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`SampleSet.fromList(samples:number[])`
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Creates a sample set distribution using an array of samples.
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<SquiggleEditor defaultCode="SampleSet.fromList([1,2,3,4,6,5,5,5])" />
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### Arguments
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- `samples`: An array of at least 5 numbers.
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<Admonition type="caution" title="Caution!">
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<p>
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Samples are converted into{" "}
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<a href="https://en.wikipedia.org/wiki/Probability_density_function">PDF</a>{" "}
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shapes automatically using{" "}
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<a href="https://en.wikipedia.org/wiki/Kernel_density_estimation">
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kernel density estimation
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</a>{" "}
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and an approximated bandwidth. Eventually Squiggle will allow for more
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specificity.
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</p>
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</Admonition>
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## PointSet.makeContinuous
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`PointSet.makeContinuous(points:{x: number, y: number})`
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Creates a continuous point set distribution using a list of points.
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<SquiggleEditor
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defaultCode={`PointSet.makeContinuous([
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{ x: 0, y: 0.1 },
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{ x: 1, y: 0.2 },
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{ x: 2, y: 0.15 },
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{ x: 3, y: 0.1 }
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])`}
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/>
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### Arguments
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- `points`: An array of at least 3 coordinates.
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## PointSet.makeDiscrete
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`PointSet.makeDiscrete(points:{x: number, y: number})`
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Creates a discrete point set distribution using a list of points.
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<SquiggleEditor
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defaultCode={`PointSet.makeDiscrete([
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{ x: 0, y: 0.1 },
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{ x: 1, y: 0.2 },
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{ x: 2, y: 0.15 },
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{ x: 3, y: 0.1 }
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])`}
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/>
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### Arguments
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- `points`: An array of at least 1 coordinate.
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