299 lines
10 KiB
Plaintext
299 lines
10 KiB
Plaintext
---
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title: "Creating Distributions"
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sidebar_position: 8
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---
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import TOCInline from "@theme/TOCInline";
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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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<TOCInline toc={toc} maxHeadingLevel={2} />
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## To
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`(5thPercentile: float) to (95thPercentile: float)`
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`to(5thPercentile: float, 95thPercentile: float)`
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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>
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<TabItem value="ex1" label="5 to 10" default>
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When `5 to 10` is entered, both numbers are positive, so it generates a
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lognormal distribution with 5th and 95th percentiles at 5 and 10.
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<SquiggleEditor initialSquiggleString="5 to 10" />
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</TabItem>
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<TabItem value="ex3" label="to(5,10)">
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`5 to 10` does the same thing as `to(5,10)`.
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<SquiggleEditor initialSquiggleString="to(5,10)" />
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</TabItem>
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<TabItem value="ex2" label="-5 to 5">
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When `-5 to 5` is entered, there's negative values, so it generates a normal
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distribution. This has 5th and 95th percentiles at 5 and 10.
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<SquiggleEditor initialSquiggleString="-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. If this
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happens, you can click the "log x scale" box to view this using a log scale.
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<SquiggleEditor initialSquiggleString="1 to 10000" />
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</TabItem>
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</Tabs>
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### Arguments
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- `5thPercentile`: Float
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- `95thPercentile`: Float, 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?: float[])`
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`mx(...distributions: Distribution[], weights?: float[])`
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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>
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<TabItem value="ex1" label="Simple" default>
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<SquiggleEditor initialSquiggleString="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 initialSquiggleString="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 initialSquiggleString="mixture(1 to 5, 8 to 10, 1, 3, 20)" />
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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 floats, each passed as a paramater. Floats will be converted into Delta distributions.
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- `weights`: An optional array of floats, 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 assignment. 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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<SquiggleEditor
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initialSquiggleString={`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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<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 assignment. I think I have an 80% chance of doing it.
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</p>
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<SquiggleEditor
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initialSquiggleString={`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:float, standardDeviation:float)`
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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>
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<TabItem value="ex1" label="normal(5,1)" default>
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<SquiggleEditor initialSquiggleString="normal(5, 1)" />
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</TabItem>
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<TabItem value="ex2" label="normal(100000000000, 100000000000)">
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<SquiggleEditor initialSquiggleString="normal(100000000000, 100000000000)" />
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</TabItem>
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</Tabs>
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### Arguments
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- `mean`: Float
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- `standard deviation`: Float 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: float, sigma: float)`
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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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<SquiggleEditor initialSquiggleString="lognormal(0, 0.7)" />
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### Arguments
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- `mu`: Float
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- `sigma`: Float greater than zero
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[Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution)
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### Argument Alternatives
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`Mu` and `sigma` can be difficult to directly reason about. Because of this complexity, we recommend typically using the <a href="#to">to</a> syntax.
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<details>
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<summary>❓ Understanding <bold>mu</bold> and <bold>sigma</bold></summary>
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<p>
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The log of `lognormal(mu, sigma)` is a normal distribution with mean `mu` and standard deviation `sigma`. For example, these two distributions are identical:
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</p>
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<SquiggleEditor
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initialSquiggleString={`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:float, high:float)`
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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 initialSquiggleString="uniform(3,7)" />
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### Arguments
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- `low`: Float
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- `high`: Float 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 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.
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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 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.
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</p>
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</Admonition>
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## Beta
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``beta(alpha:float, beta:float)``
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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>
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<TabItem value="ex1" label="beta(10, 20)" default>
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<SquiggleEditor initialSquiggleString="beta(10,20)" />
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</TabItem>
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<TabItem value="ex2" label="beta(1000, 1000)" >
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<SquiggleEditor initialSquiggleString="beta(1000, 2000)" />
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</TabItem>
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<TabItem value="ex3" label="beta(1, 10)" >
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<SquiggleEditor initialSquiggleString="beta(1, 10)" />
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</TabItem>
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<TabItem value="ex4" label="beta(10, 1)" >
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<SquiggleEditor initialSquiggleString="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 initialSquiggleString="beta(0.8, 0.8)" />
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</TabItem>
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</Tabs>
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### Arguments
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- `alpha`: Float greater than zero
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- `beta`: Float 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 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.
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</p>
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<details>
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<summary>Examples</summary>
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<Tabs>
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<TabItem value="ex1" label="beta(0.3, 0.3)" default>
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<SquiggleEditor initialSquiggleString="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 initialSquiggleString="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 initialSquiggleString="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 initialSquiggleString="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:float)``
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Creates an [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution) with the given rate.
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<SquiggleEditor initialSquiggleString="exponential(4)" />
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### Arguments
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- `rate`: Float greater than zero
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## Triangular distribution
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``triangular(low:float, mode:float, high:float)``
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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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#### Validity
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### Arguments
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- `low`: Float
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- `mode`: Float greater than `low`
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- `high`: Float greater than `mode`
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<SquiggleEditor initialSquiggleString="triangular(1, 2, 4)" />
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## FromSamples
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Creates a sample set distribution using an array of samples.
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<SquiggleEditor initialSquiggleString="fromSamples([1,2,3,4,6,5,5,5])" />
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#### Validity
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For `fromSamples(xs)`,
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- `xs.length > 5`
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- Strictly every element of `xs` must be a number.
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