Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Better-than-average uniform random variables and Eulerian numbers, or: How many candidates should a voter approve?

Svante Janson, Warren D. Smith

Abstract

Consider $n$ independent random numbers with a uniform distribution on $[0,1]$. The number of them that exceed their mean is shown to have an Eulerian distribution, i.e., it is described by the Eulerian numbers. This is related to, but distinct from, the well known fact that the integer part of the sum of independent random numbers uniform on $[0,1]$ has an Eulerian distribution. One motivation for this problem comes from voting theory.

Better-than-average uniform random variables and Eulerian numbers, or: How many candidates should a voter approve?

Abstract

Consider independent random numbers with a uniform distribution on . The number of them that exceed their mean is shown to have an Eulerian distribution, i.e., it is described by the Eulerian numbers. This is related to, but distinct from, the well known fact that the integer part of the sum of independent random numbers uniform on has an Eulerian distribution. One motivation for this problem comes from voting theory.
Paper Structure (5 sections, 4 theorems, 39 equations)

This paper contains 5 sections, 4 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs
  4. Application in (or motivation from) voting theory
  5. A probability for exponential variables

Key Result

Theorem 1.1

Define $W_n$ by wn, for some $n\geqslant2$. Then, for every $k\geqslant0$,

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • proof : Proof of Theorem
  • ...and 5 more