The Inspection Paradox and Length-Biased Sampling

An exchange publishes that the average gap between trades on a thinly-traded bond is 10 seconds. You connect at a random moment and start timing the gap you land inside of — and it averages out to nearly 20 seconds, almost double the published figure. Neither number is wrong; you've stumbled into the inspection paradox. When you arrive at a random time rather than picking a random gap from a list, you're more likely to land inside a long gap than a short one, simply because long gaps occupy more of the timeline. Your random arrival isn't sampling gaps uniformly — it's sampling them weighted by their own length, a subtly different and systematically biased process called length-biased sampling.

An analogy: waiting for a bus with irregular intervals

Buses are scheduled to arrive on average every 10 minutes, but with irregular gaps — some 2 minutes apart, some 20. If you show up at a uniformly random time and ask "how long is the gap between the bus before me and the bus after me," you'll disproportionately land inside the long 20-minute gaps, because they cover twice as much clock-time as the short 2-minute gaps, even though there might be an equal count of each. A passenger who tracks every individual gap and averages them (treating each gap equally, regardless of length) gets the true average of 10 minutes. A passenger who just shows up once and measures whatever gap they land in gets a number biased upward — often close to double the true average, exactly because long gaps swallow more calendar time and are more likely to be "the one you're in" at any given moment.

The logic, one piece at a time

Let intervals between events have true average length $\mathbb{E}[T]$ (this is the "count each gap once" average). If instead you sample by picking a uniformly random time point and asking "how long is the interval containing this point," the resulting length-biased distribution weights each interval by its own length divided by the total, giving an expected value of

$$\mathbb{E}[T_{\text{observed}}] = \frac{\mathbb{E}[T^2]}{\mathbb{E}[T]} = \mathbb{E}[T] + \frac{\text{Var}(T)}{\mathbb{E}[T]}.$$

In plain English: the length you observe by landing at a random moment equals the true average interval plus an extra term that grows with how variable the intervals are — the more spread out interval lengths are, the bigger the gap between "true average" and "what a random arrival sees." If all intervals were exactly equal (zero variance), there'd be no paradox at all; the paradox is entirely a symptom of variability in interval length.

Worked example 1: two interval lengths

Suppose trade gaps alternate, with equal counts, between 2 seconds and 18 seconds. The true average (each gap counted once) is $\mathbb{E}[T] = (2+18)/2 = 10$ seconds. But time-weighted, the 18-second gaps occupy $18/(2+18) = 90\%$ of total elapsed time versus only $10\%$ for the 2-second gaps. A random-time arrival lands in an 18-second gap 90% of the time and a 2-second gap 10% of the time, giving $\mathbb{E}[T_{\text{observed}}] = 0.9 \times 18 + 0.1 \times 2 = 16.2 + 0.2 = 16.4$ seconds — 64% higher than the true 10-second average, purely from the length-biasing of random-time sampling.

Worked example 2: the exponential case, matching the opening claim

For exponentially distributed gaps with mean $\mathbb{E}[T] = 10$ seconds (a common model for trade arrivals), the variance equals the mean squared, $\text{Var}(T) = 100$, so $\mathbb{E}[T_{\text{observed}}] = 10 + 100/10 = 10 + 10 = 20$ seconds — exactly double the true mean. This is a famous special case: for exponential (memoryless) inter-arrival times, random-time inspection always doubles the apparent average gap, regardless of the actual rate, which matches the opening claim of the published 10-second average looking like 20 seconds under casual observation.

The exponential density above has most of its mass near small values but a long tail of rare, large gaps — those rare large gaps are exactly the ones that dominate a random-time snapshot, which is why the doubling result in worked example 2 holds for any exponential rate.

What this means in practice

The inspection paradox shows up whenever you measure "the interval you happen to be in" rather than sampling intervals directly: apparent trade-gap statistics gathered by polling at fixed intervals, perceived queue wait times (you're more likely to arrive during a busy, slow-moving period than a fast one), and any metric computed by "snapshot" sampling of an ongoing process with variable-length episodes. The fix is to either sample entire episodes directly (count each gap once, regardless of when you happened to look) or explicitly correct with the length-biasing formula above when only random-time snapshots are available.