Size bias for one and all

TL;DR

This paper reviews the concept of size bias, exploring its occurrence in various mathematical and statistical contexts, including paradoxes, sampling, and distribution analysis, highlighting its surprising connections.

Contribution

It provides a comprehensive survey of size bias and its unexpected links across multiple areas of probability and number theory.

Findings

Size bias appears in waiting-time paradoxes and sampling schemes.

Connections between size bias and Stein's method, distribution analysis, and number theory.

The paper offers a unified overview of these diverse applications.

Abstract

Size bias occurs famously in waiting-time paradoxes, undesirably in sampling schemes, and unexpectedly in connection with Stein's method, tightness, analysis of the lognormal distribution, Skorohod embedding, infinite divisibility, and number theory. In this paper we review the basics and survey some of these unexpected connections.