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Determinity Distribution

A Probability Distribution with Events' Frequency Range Guarantees.
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When dealing with problems, like aging, or playing games, like Tetris, it is tempting to use probability distribution over the intake materials or falling bricks. However, just having a probability distribution does not guarantee, that a specific type of falling brick will ever occur, it just says that it's highly unlikely for it to not occur.

To solve for infinite lifespans, or infinitely long playing of Tetris, what's needed, is a guarantees of the frequencies, or even patterns. Hence, the idea of a "Determinity Distribution" or "Deterministic Probability Distribution", which sounds like an oxymoron, but could be defined simply.

Suppose we have an algebra of events (e.g., like when defining a random variable), and add a requirement, that the events have to repeat at no rarer or more often than threshold frequencies. That automatically implies that only specific periodic or aperiodic patterns of event can qualify.

Surely, you may think a periodic pattern already has a "Determinity Distribution", because we have frequency guarantees. However, it is not a probability distribution, because we already know the pattern with certainty. However, given the inequalities (e.g., thresholds, or upper / lower bounds) for frequencies, it may be possible to construct many patterns, and even aperiodic patterns (just like aperiodic fractions, like irrational numbers with proofs), that guarantee those frequencies, and at the same time are not exactly predictable (!).

Thus, a determinity distribution would define an equivalence class for all patterns, that the constraints of the probability distribution with frequency guarantees.

This also may be applicable to lottery games. For example, lottery makers may claim specific probabilities about their games, but not guarantee them, so that the winning prize actually is never really won. However, giving legal requirements for determinity distribution would change the games... And yes, determinity distribution would imply, that the more observations we have, the more certain other events are, but not completely certain (until the last event before threshold). It's kind of an interesting concept mathematically, that you can have such interplay of indeterminism and determinism.

Inyuki, Oct 20 2019

Shapeflow Shapeflow
(the idea that inspired this) [Inyuki, Oct 20 2019]


       Is this just a kind of Poisson distribution?
pocmloc, Oct 20 2019

       I thought the Poisson distribution described the distribution of negative votes on the HB.
MaxwellBuchanan, Oct 20 2019

       I was thinking a truncated gaussian. If you only allow n (or if you're feeling more specific, a couple of) standard deviations either side, you're effectively guaranteeing every possibility in the range, given a large (or small) enough ratio between options, and sample size. If you're talking big permutational combinations, that number can get so big, you'll never fetch every individual item, even if you start at the beginning and aim to work your way right to the end with a probability of 1 for each.   

       But I'm confused about the idea of guaranteeing a given outcome - given enough time, space and chances, Murphy's Law ought to apply - which is as close to a guarantee as you're likely to get. With enough options on the table, it may well take until the end of the universe for a given number to come up. Certainly longer than any individual's lifespan - so guarantees seem like they'd be tricky to offer.   

       You could define some pseudo-random sequence, or just pick a particular permutation that covered the entire range given enough goes - guaranteeing hitting everything seems like another way of saying you'll avoid double-picks (or for a softer version, limiting them in some way) - and this eventually means narrowing down the list of possibles so much that the odds would be shortened considerably towards the end of a run.
zen_tom, Oct 20 2019


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