Section 12.4

The Inspection Paradox

Length-biased sampling: the wait for a bus, the arc that contains a fixed point.

Toolkit

A fixed observer or point does not land in a “typical” interval of a renewal/partition process; it lands in one chosen with probability proportional to its own length. This size-biasing systematically inflates the relevant expectation above the naive mean.

Key facts:

\mathbb{P}(\text{point falls in an interval of length } \ell) \propto \ell \cdot f(\ell),

so the length of the interval containing the point has size-biased density $f^*(\ell)=\ell f(\ell)/\mathbb{E}[L]$ , giving

\mathbb{E}[L^*] = \frac{\mathbb{E}[L^2]}{\mathbb{E}[L]} \ge \mathbb{E}[L].

For a renewal process with i.i.d. gaps $L$ , the forward (or backward) recurrence time from a fixed time point has mean

\mathbb{E}[\text{wait}] = \frac{\mathbb{E}[L^2]}{2\,\mathbb{E}[L]},

equal to $\mathbb{E}[L]/2$ only when $L$ is deterministic.

For points scattered independently and uniformly, conditioning on the covering arc and splitting it at the fixed point reduces the two pieces to minima of independent uniforms, each with the usual order-statistic expectation.

Problems