Poisson Processes: Arrivals, Thinning, and the Inspection Paradox

The distribution zoo introduced Poisson counts and exponential waits; the Poisson process is the machine that generates both — the default model for anything arriving randomly in time: orders, trades, news, defaults, support tickets. Its closure properties and one famous paradox supply a steady stream of interview questions.

The definition and the dictionary

A Poisson process at rate $\lambda$ has independent increments, with the count in any window of length $t$ distributed Poisson($\lambda t$), and gaps between arrivals iid Exponential($\lambda$). Counts and gaps are two views of one object — pick whichever makes the question easy. Memorylessness of the gaps (the zoo lesson) means the process has no clock: at any moment, expected time to the next arrival is $1/\lambda$, regardless of history.

The closure properties that solve problems

• Superposition: merge independent Poisson streams (buys at rate $\lambda_B$, sells at rate $\lambda_S$) → Poisson at $\lambda_B + \lambda_S$. First-of-many races: the next event is a buy with probability $\lambda_B / (\lambda_B + \lambda_S)$ — instantly answers "three counters serve at rates 1, 2, 3 per minute; who serves you first?"
• Thinning: keep each arrival independently with probability $p$ → Poisson at $p\lambda$. Trades at 100/sec, 2% are informed → informed flow is Poisson at 2/sec, independent of the uninformed stream — the assumption structure of the microstructure course's models, now with a name.
• Conditional uniformity: given $n$ arrivals happened in $[0, T]$, their times are distributed like $n$ iid uniforms — order statistics, connecting directly to that lesson. "Given 5 trades this minute, expected time of the third?" — $\frac{3}{6}$ of the minute: 30 seconds.

The inspection paradox

Buses arrive as a Poisson process, one per 10 minutes on average. You show up at a random time. Expected wait?

Intuition says 5 minutes (half the average gap). The truth is 10 minutes — memorylessness means your arrival learns nothing, and the full $1/\lambda$ remains. The resolution of the apparent contradiction: the interval you land in is not a typical interval — long gaps cover more of the timeline, so random arrivals disproportionately fall into them (size-biased sampling; for exponential gaps, the straddling interval averages $2/\lambda$, and you wait half of it).

This paradox is a genre, not a curiosity: "average class size experienced by students exceeds average class size on the books," "your friends have more friends than you do," "the average fund investors are in underperforms the average fund" — all size-biasing. When any question mixes "average over items" with "average over encounters," flag the two averages before computing either; that flag is usually the whole point.

Beyond constant rates — the honest caveat

Real arrival streams (trades, especially) violate constant-$\lambda$: intensity has a daily U-shape and clusters (activity begets activity). Inhomogeneous Poisson handles the first; Hawkes processes — where each arrival temporarily raises the arrival rate — handle the second and have become the standard microstructure model of order flow. For interviews, one sentence suffices: "I'd start Poisson, then check for clustering; trade arrivals are famously self-exciting." It shows the model and its expiration date, the pattern this curriculum keeps rewarding.

The interview version

"A desk gets calls at rate 2/hour, 30% are urgent. Expected time until the second urgent call?" — Thin to rate 0.6/hour; gamma/Erlang wait: $2/0.6 \approx 3.3$ hours. "You walk onto the floor at a random moment; how long since the last trade in a name trading once per minute?" — one full minute backward and one forward in expectation (memorylessness both directions) — and if that feels wrong, you've rediscovered the inspection paradox, which is exactly the discussion the interviewer wanted.