How to Scale a Pareto Distribution Between 0 and 1?

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Discussion Overview

The discussion revolves around the challenge of scaling a Pareto distribution to fit within the range of 0 to 1. Participants explore the implications of modifying the distribution and consider alternative approaches to achieve a bounded distribution suitable for number generation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant seeks guidance on generating numbers based on a Pareto distribution and expresses the need to scale the distribution to fit between 0 and 1.
  • Another participant points out that the traditional Pareto distribution does not start at zero due to a singularity in its probability density function (PDF) and questions whether the entire range should be rescaled or if a Pareto-like distribution is desired.
  • A later reply suggests that if a Pareto-like distribution is intended, it may be necessary to define a bounded version of the distribution and calculate the cumulative distribution function (CDF) to transform a uniform variable into the desired distribution.
  • Further clarification indicates that a bounded Pareto distribution could be appropriate, but emphasizes that the choice of the original range will significantly affect the resulting distribution's shape.

Areas of Agreement / Disagreement

Participants generally agree on the need for a bounded distribution but have differing views on how to achieve this and the implications of scaling the Pareto distribution. The discussion remains unresolved regarding the specific parameters and methods to use.

Contextual Notes

Limitations include the need for clarity on the desired probability distribution and the effects of scaling on the distribution's characteristics. The discussion does not resolve the mathematical steps required to achieve the transformation.

jianxu
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Hello and thank you for taking the time to read this.

I am making a number generator that generates a number based on a pareto distribution.
The problem is, the distribution essentially goes from 0 to infinity. How would I go about scaling the values so I get a range between 0 and 1 instead?

As of right now, I'm taking a random uniform number and simply turning it into a pareto distribution using a method called Generating generalized Pareto random variables (found in wikipedia).

Thanks again!
 
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I think you need to specify more exactly what kind of probability distribution you want. The Pareto distribution doesn't start at zero because the PDF has a singularity there, so it can't integrate to 1.

Do you want the whole Pareto range rescaled onto the interval from 0 to 1? If you do that, it won't be a Pareto distribution anymore by any stretch of the imagination.

Or do you want a Pareto-like power law distribution that goes from 0 to 1? Once you have decided which pdf you want (e.g.
a bounded Pareto distribution shifted/scaled to start at zero), that should be fairly straightforward, you should be able to calculate the CDF and invert it - which you can use to transform a uniform variable into your distribution.
 
Hello,

Thanks for the reply and sorry for the confusion.
I think a pareto like distribution from 0 to 1 is what I'm going for.
So would I just have it be bounded between 0 and 1?
Thanks!
 
jianxu said:
Hello,

Thanks for the reply and sorry for the confusion.
I think a pareto like distribution from 0 to 1 is what I'm going for.
So would I just have it be bounded between 0 and 1?
Thanks!
In that case the bounded Pareto distribution (also described on the wikipedia page) is probably what you are looking for. That is only defined for ranges that start at values greater than zero though, so you'll have to pick some range and then linearly scale/shift the values to your desired range. Which range you pick for the original bounded Pareto distribution will have quite a significant effect on the shape of the resulting distribution, so you'll need to find a reasonable set of parameters for whatever you're modelling.
 

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