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arxiv: 2606.30287 · v1 · pith:JFTRCO7Knew · submitted 2026-06-29 · 🧮 math.PR · math.CO

On the Probability a Weighted Bernoulli Sum Exceeds Its Mean

Aleksa Milojevic , Benny Sudakov This is my paper

Pith reviewed 2026-06-30 05:13 UTC · model grok-4.3

classification 🧮 math.PR math.CO
keywords weighted Bernoulli sumprobability above meanManickam-Miklós-Singhi conjectureBernoulli random variablesexpectation inequalitycombinatorial conjecturetail bound
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The pith

For weights summing to 1, a weighted sum of i.i.d. Bernoulli(p) trials satisfies P[X ≥ E[X]] ≥ p whenever p ≤ 1/3.

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

The authors conjecture that any convex combination X of independent Bernoulli random variables with common success probability p has at least probability p of landing at or above its expectation, provided p is at most one third. Establishing this would give a simple, uniform lower bound on the mass above the mean for this family of random variables. They show that a combinatorial conjecture of Manickam, Miklós and Singhi implies the probability statement for all sufficiently small p, thereby proving the claim in that regime.

Core claim

Let w1 to wm be positive reals summing to 1 and let v1 to vm be independent Bernoulli(p) random variables. Let X be the weighted sum sum wi vi. The paper conjectures that P(X ≥ E[X]) ≥ p for every p between 0 and 1/3 inclusive. It further observes that a suitable form of the Manickam-Miklós-Singhi conjecture is sufficient to prove the probability lower bound when p is small enough.

What carries the argument

The link between the probability inequality and a version of the Manickam-Miklós-Singhi conjecture, which is used to prove the bound for small p.

If this is right

  • The conjectured bound holds for all sufficiently small p.
  • If the invoked MMS statement is true, then the probability inequality holds for all p ≤ 1/3.
  • The result applies uniformly to any choice of positive weights summing to one.

Where Pith is reading between the lines

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

  • The bound might extend to p larger than 1/3 for some weights, though this is not addressed.
  • Similar inequalities could be studied for other distributions beyond Bernoulli.
  • Numerical checks for moderate p could provide supporting evidence or counterexamples.

Load-bearing premise

The version of the Manickam-Miklós-Singhi conjecture invoked holds in the needed form for the small-p regime.

What would settle it

A counterexample consisting of specific weights summing to 1 and a value of p at most 1/3 where the probability P[X ≥ E[X]] falls below p would disprove the conjecture.

read the original abstract

Let $w_1, \dots, w_m$ be positive real weights whose sum is $1$, and let $v_1, \dots, v_m$ be i.i.d. Bernoulli$(p)$ random variables. If we let $X=\sum_{i=1}^m w_i v_i$, then we conjecture that for all $0\leq p\leq 1/3$ we have \[\mathbb{P}\big[X\geq \mathbb{E}[X]\big]\geq p.\] In this short note, we observe a connection of this conjecture with a version of the Manickam-Mikl\'os-Singhi conjecture, which allows one to prove it for sufficiently small values of $p$.

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

1 major / 0 minor

Summary. The manuscript conjectures that for positive weights w_1 to w_m summing to 1 and X the corresponding weighted sum of i.i.d. Bernoulli(p) random variables, P(X ≥ E[X]) ≥ p holds for all 0 ≤ p ≤ 1/3. It further observes that a connection to a version of the Manickam-Miklós-Singhi conjecture yields a proof of the bound for all sufficiently small p.

Significance. If established, the conjectured bound would give a uniform, parameter-free lower bound on the probability that a weighted Bernoulli sum meets or exceeds its mean, with potential implications for concentration and extremal problems in probability. The explicit link to the MMS conjecture is a constructive observation that correctly reduces the small-p regime to an existing combinatorial statement; this partial result is presented as conditional rather than unconditional.

major comments (1)
  1. [the paragraph discussing the connection to the Manickam-Miklós-Singhi conjecture] The abstract and the note on the MMS connection assert that the link 'allows one to prove' the bound for sufficiently small p, yet the manuscript provides neither the explicit derivation nor the precise form of the MMS conjecture invoked. This is load-bearing for the partial result claimed in the abstract, as the reader cannot verify the reduction from the given text alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting this point about the presentation of the MMS connection. We address the major comment below.

read point-by-point responses

  1. Referee: The abstract and the note on the MMS connection assert that the link 'allows one to prove' the bound for sufficiently small p, yet the manuscript provides neither the explicit derivation nor the precise form of the MMS conjecture invoked. This is load-bearing for the partial result claimed in the abstract, as the reader cannot verify the reduction from the given text alone.

    Authors: We agree that the manuscript does not currently supply the explicit derivation or the precise statement of the version of the Manickam-Miklós-Singhi conjecture invoked. The reduction is therefore not verifiable from the text as written. In the revised version we will add a self-contained paragraph (or short section) that states the relevant form of the MMS conjecture and outlines the steps by which it yields the claimed bound for all sufficiently small p. This change will make the partial result fully checkable while preserving the note's brevity and focus. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states a probability conjecture for all p ≤ 1/3 and observes that a version of the external Manickam-Miklós-Singhi conjecture implies the bound only for sufficiently small p. No equations or steps reduce the target probability bound to a fitted parameter, self-definition, or self-citation chain. The partial result is explicitly conditional on an independent combinatorial conjecture whose authors do not overlap with the present paper, and the full statement remains open. This satisfies the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The partial proof depends on the validity of a specific version of the Manickam-Miklós-Singhi conjecture being applicable to the weighted Bernoulli setting for small p; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption A version of the Manickam-Miklós-Singhi conjecture holds and applies to prove the probability bound for sufficiently small p
    The abstract states that this connection allows one to prove the conjecture for sufficiently small values of p.

pith-pipeline@v0.9.1-grok · 5645 in / 1184 out tokens · 34908 ms · 2026-06-30T05:13:50.931486+00:00 · methodology

discussion (0)

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Reference graph

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