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arxiv: 2506.12369 · v4 · submitted 2025-06-14 · 🧮 math.PR

Tossing half-coins and other partial coins: signed probabilities and Sibuya distribution

Nikolai Leonenko , Igor Podlubny This is my paper

Pith reviewed 2026-05-19 09:45 UTC · model grok-4.3

classification 🧮 math.PR
keywords signed probabilitiesfractional coin tossesSibuya distributionnumerical simulationsigned measuresrandom variate generationprobability distributions
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The pith

Tossing one-nth of a coin generates samples from signed probability distributions numerically.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a numerical method for simulating signed probability distributions by treating the toss of a 1/n-th coin as the core operation. This technique is illustrated with concrete examples and connects the outputs to the Sibuya distribution. A sympathetic reader would value the approach because it supplies a practical way to generate random samples from measures that can take negative values, something standard positive-probability simulators cannot handle directly. The method aims to keep the generated results aligned with the axioms that govern signed measures.

Core claim

The authors introduce a simulation procedure for signed probability distributions that arises when a coin is conceptually divided into n equal parts and only one part is tossed; the procedure produces random variates whose statistics remain consistent with the rules of signed measures and is demonstrated through explicit examples involving the Sibuya distribution.

What carries the argument

The numerical simulation algorithm that converts fractional coin tosses into random samples drawn from a signed measure via the Sibuya distribution.

If this is right

  • The outputs remain consistent with the axioms of signed measures for any chosen fraction 1/n.
  • Random variates can be produced for distributions that include negative probability masses.
  • The same procedure applies across a range of fractional coin sizes without changing the underlying logic.
  • Concrete numerical examples confirm that the generated samples match the expected behavior of the Sibuya distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The simulation could be tested by checking whether sample moments converge to the analytically known values of the Sibuya distribution as sample size grows.
  • The technique may connect to generating functions used for other discrete distributions that allow signed weights.
  • Extending the method to continuous-time analogs of fractional coin tosses could link it to fractional Poisson processes.

Load-bearing premise

Signed probability distributions that arise from tossing one-nth of a coin admit a stable numerical simulation whose outputs stay consistent with signed-measure axioms without requiring extra validation or error bounds.

What would settle it

Generate a large sample using the method for a fixed n, compute the empirical signed probabilities or moments, and compare them to the known closed-form values of the Sibuya distribution for that n; systematic mismatch would show the simulation does not preserve the required properties.

Figures

Figures reproduced from arXiv: 2506.12369 by Igor Podlubny, Nikolai Leonenko.

Figure 1
Figure 1. Figure 1: Coefficients of g(x) and f(x) 6. Numerical simulations 6.1. Evaluation of the coefficients of f(x) and g(x). First, let us consider the generating function of the Sibuya distribution (4). The coeffi￾cients of the power series expansion of g(x) for 0 < α < 1 are all positive: (20) gk(x) =  1 − (1 − x) α  x k = X∞ n=1 w˜ (α) n x n , w˜ (α) n > 0, n = 1, 2, 3, . . . , where (21) ˜w (α) n = (−1)n+1 α n  , … view at source ↗
Figure 2
Figure 2. Figure 2: A quarter-coin. Flips: 10000; ones: 1183; zeros: 8817; expectation: 0.1183 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A one-third-coin. Flips: 10000; ones: 1650; zeros: 8350; expectation: 0.1650 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A half-coin. Flips: 10000; ones: 2519; zeros: 7481; expectation: 0.2519 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A three-quarters-coin. Flips: 10000; ones: 3716; zeros: 6284; expectation: 0.3716 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A four-fifths-coin. Flips: 10000; ones: 3997; zeros: 6003; expectation: 0.3997 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A five-sixths-coin. Flips: 10000; ones: 4213; zeros: 5787; expectation: 0.4213 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Two half-coins. Flips: 20000; ones on both partial coins: 1200; ones on one partial coin: 7525; zeros on both partial coins: 11275; expectation: 0.4962 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A one-third-coin and a two-thirds-coin. Flips: 20000; ones on both partial coins: 1029; ones on one partial coin: 7793; zeros on both partial coins: 11178; expectation: 0.4925 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A half-coin and a two-thirds-coin. Flips: 20000; ones on both partial coins: 1624; ones on one partial coin: 8357; zeros on both partial coins: 10019; expectation: 0.5802 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A biased half-coin with a = 0.4 and b = 0.6. Flips: 10000; ones: 9156; zeros: 844; expectation: 0.9156 [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: A biased half-coin with a = 0.4 and b = 0.6. Flips: 10000; ones: 824; zeros: 9176 ; expectation: 0.0824 [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: A biased three-quarters-coin with a = 0.7 and b = 0.3. Flips: 10000 ; ones: 847; zeros: 9153; expectation: 0.0847 [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: September 13th, 2024, room M13, Isaac Newton Institute, Cambridge: the first successful simulation of a half-coin. (SSD), supported by EPSRC grant EP/Z000580/1, at Isaac Newton Institute for Math￾ematical Sciences, Cambridge. The first successful simulation of a half-coin tossing happened during the SSD programme on September 13th, 2024, in the room M13 of the Isaac Newton Institute ( [PITH_FULL_IMAGE:fi… view at source ↗
read the original abstract

A method for the numerical simulation of signed probability distributions for the case of tossing $1/n$-th of a coin is presented and illustrated by examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a method for the numerical simulation of signed probability distributions arising from tossing a 1/n-th coin. The approach relies on the Sibuya distribution and is illustrated through explicit examples that demonstrate consistency with the axioms of signed measures.

Significance. If the simulation procedure is rigorously validated, the paper offers a concrete computational tool for handling signed probabilities in fractional coin-toss models. This could facilitate numerical studies of processes where negative probabilities arise naturally, such as certain branching or renewal models linked to the Sibuya distribution. The explicit examples provide a useful starting point for reproducibility.

minor comments (3)
  1. The abstract and introduction would benefit from a brief statement of the precise convergence criterion or error bound used to validate the simulation outputs against the signed-measure axioms.
  2. Notation for the signed probabilities (e.g., the definition of the measure on the state space) should be introduced earlier and used consistently in the example sections.
  3. A short discussion of computational cost or stability for large n would help readers assess practical applicability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The summary accurately captures the contribution, and we appreciate the note on potential utility for numerical studies involving signed measures and the Sibuya distribution.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a numerical simulation procedure for signed measures in the context of 1/n-coin tosses, grounded in the Sibuya distribution and demonstrated through concrete examples. No load-bearing step reduces by construction to a fitted parameter, self-citation, or definitional equivalence; the method is presented as an independent construction whose consistency is shown directly via the examples rather than assumed from prior fitted outputs. The derivation chain remains self-contained against the stated axioms of signed measures.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard extension of probability measures to signed measures and on the known properties of the Sibuya distribution; no new free parameters or invented entities are visible in the abstract.

axioms (2)
  • standard math Signed measures satisfy the usual additivity and normalization properties of probability measures except that total mass may be signed.
    Implicit in any treatment of signed probabilities.
  • standard math The Sibuya distribution is a well-defined discrete probability distribution on the positive integers.
    The abstract uses the Sibuya distribution as the illustrative example.

pith-pipeline@v0.9.0 · 5533 in / 1184 out tokens · 41861 ms · 2026-05-19T09:45:15.129352+00:00 · methodology

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Works this paper leans on

14 extracted references · 14 canonical work pages

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    Podlubny, I., Partial coin, 2025. MATLAB Central File Exchange, Submission No. 181284 (2025). https://www.mathworks.com/matlabcentral/fileexchange/181284 Nikolai Leonenko, School of Mathematics, Cardiff University, UK Email address : leonenkon@cardiff.ac.uk Igor Podlubny, BERG F aculty, Technical University of Kosice, Slovakia Email address : igor.podlubn...

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