Determining if a list of numbers is a result of multiplication

[alk10]

TL;DR Summary

Determining if a list of numbers is a result of multiplication

Suppose I have 2 collections of lists.

In the first collection the lists consists of random integers, with most (but not all) in the range 0-1000.
In the second collection the lists consist of integers calculated in the following way:
a. start with a random integer of similar range to the first list
b. multiply by some unknown fraction, typically (but not always) in the range 0-2.
c. round to the nearest integer

Given a particular list, I would like to be able to predict which collection it comes from.

I have tried taking the modulo from every number between 2-20 and looking at the remainder (as for example if the fraction in b) was exactly 2, then the elements mod 2 would always be zero), but couldn't find a noticeable difference. Would appreciate any ideas.

 

[FactChecker]

You have been very vague about the probabilities or distributions of the random behavior.
Phrases like "most (but not all)", "typically (but not always)", "random integers", etc. do not give us much to work with.
Since the second method might create a large number of integers that are beyond the range 0-1000, I would consider using that to make an educated guess about which method created the list. But your description is too vague to know if that is a feasible method.

 

[Dale]

I would use a Bayesian approach for this (of course). "Simply" write down the data-generating models for your two possibilities and then do a Bayesian analysis for any parameters of the models. Then you can compare the models using Bayes factors or your favorite alternative Bayesian model comparison technique.

 

[BWV]

Just run a Monte Carlo on some randomly generated numbers to test

 

[FactChecker]

In general, the distributions of the two processes will be significantly different. You should be able to use a Chi-square goodness of fit test to determine which method is more likely for a given sample. Without more information about the distributions, I don't think that much more can be said.