Abstract
In the famous “waiting time paradox”, see Feller 38, Section I.4, there are two
plausible but conflicting analyses of the waiting time for the next bus, once you
get to the bus stop. More formally, this paradox concerns the waiting time W t for
the next arrival, starting from an arbitrary instant t, in a standard homogeneous
Poisson process with intensity parameter λ = 1: (a) The lack of memory of the
exponential interarrival time suggests that EW t is not sensitive to the choice of
t; so EW t = EW 0 = 1. (b) Since the starting time is chosen uniformly in the
interval between two successive arrivals, an interval of mean length 1, symmetry
suggests that EW t = 1/2.
As Feller shows, the reasoning behind both analyses is faulty, because it is the
instant and not the interval which is arbitrary: a longer interval thereby becomes
more likely than the relative frequencies of interarrival lengths would suggest, a
canonical instance of size biasing. So an unqualified appeal to properties of the
original interarrival distribution is fallacious.
In fact, as we will discuss, a reasonable but precise interpretation of “arbitrary
instant” leads to the answer given in (a), though not for the reason given in (a).
Not just recreational chestnuts, but also practical matters, such as statistical
sampling tasks, are bedeviled by size bias; we provide a few references later.
Surprisingly, however, size bias plays a role in such unexpected contexts as
Stein’s method, Skorohod embedding, nonuniqueness in the method of moments,
infinite divisibility of distributions, branching processes, and number theory. We
will return to the “paradox” shortly, after giving the basics of size bias. Then
we will survey size bias as it appears in some of the non-sampling contexts. 1
In 7, pp. 78–80, the authors introduce their two and one half page survey of
size bias by saying “Size-biasing arises naturally in statistical sampling theory
(cf. Hansen and Hurwitz (1943) 46, Midzuno (1952) 66 and Gordon (1993)
44), and the results we present below are all well known in the folk literature.”
In the present paper, we feel that we have contributed a number of new results:
the conceptual heuristic given in Section 3 to explain (26), where a sum of inde-
pendent variables is size biased by biasing only a single term, the explanation of
an intimate connection between uniform integrability and tightness in Section
8, the size bias perspective on Skorohod embedding in 10, and the treatment
of infinite divisibility in Section 11 — at least the argument based on (85), size
biasing a sum by size biasing a single summand.
Another survey of size bias, with a different focus, is 24.
