Exercise 3: Overdispersion in a Poisson-Modeled Metric — Possible Solution
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WHAT THE OBSERVATION MEANS
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Per this chapter's own defining property, a genuinely Poisson-
distributed metric must have its variance exactly equal to its mean
(Var(X) = E[X] = lambda). If a team observes the real variance
consistently running much HIGHER than the observed mean, that is a
direct violation of this chapter's own mean-equals-variance property -
this metric is not behaving like a true Poisson process. This
specific pattern is called overdispersion.

WHY OVERDISPERSION HAPPENS
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Per this chapter's own overdispersion finding, the most common real
cause is that the events are not actually independent, which is one
of the Poisson distribution's own required assumptions (inherited
from its relationship to the binomial distribution's own independence
requirement, Chapter 3's own territory). If incidents tend to cluster
- a single root cause (a bad deploy, a shared dependency outage)
triggering several incidents together in a short window, rather than
each incident occurring purely independently at random - the variance
of the observed counts increases well beyond what pure Poisson
randomness alone would produce, since clustered events push some
weeks toward "many incidents at once" and others toward "few or none"
far more than independent randomness would.

WHY A PURE POISSON MODEL IS NOW MISLEADING
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Since capacity and reserve planning built on the Poisson model
(exactly this chapter's own weekly-incident-cost-style reasoning)
assumes the mean alone is enough to characterize how spread out real
outcomes will be (because variance is supposed to equal the mean), an
overdispersed real process will produce genuinely extreme weeks (very
many incidents at once) far more often than the model predicts. A
team planning reserves or on-call staffing based on the Poisson
model's own (too-narrow) variance would be systematically under-
prepared for the real clustering behavior actually happening -
exactly the same kind of "the model understates extreme outcomes"
danger Chapter 6's own independence warning raised for the binomial
distribution, now showing up in this chapter's own distribution
instead.

WHY THIS WORKS AS AN ANSWER
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The explanation is grounded directly in this chapter's own mean-
equals-variance property and its own named overdispersion finding,
identifying broken independence (clustered root causes) as the
specific mechanism, and connects the practical consequence back to
capacity/reserve planning rather than treating "the model is wrong"
as a standalone, unexplained conclusion.