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Chapter 4

Interval scaling and sums — transforming λ across intervals

Convert granularity and read combined counts. λ scales proportionally with the interval, and independent Poisson distributions combine by summing their λ.

λ applies across intervals

When you reach for Poisson, the first instinct should be to transform a reference rate into the window you care about. And when two streams are independent, you can simply add their λ values together.

rate × interval30 min → 60 minindependent sumλ adds

Recall from Chapter 3 that the mean and the variance both equal λ. So changing the interval changes λ, and that automatically moves the mean and the variance together. Throughout this chapter, every "λ moves" is also "the mean and the variance move with it."

30 min / λ = 3 30 min / λ = 2 60 min / λ = 5

Viewing two independent 30-minute windows as a single 60-minute window, λ adds to 5.

Change the window, and λ scales proportionally

The Poisson mean count λ is proportional to the length of the observation window. If the rate is 0.5 events per minute on average, then 2 minutes gives 1, and 10 minutes gives 5.

λ_new = λ_base × (new interval / base interval)
  • λ = 6 per hour → λ = 3 per 30 min (halved)
  • λ = 6 per hour → λ = 12 per 2 hours (doubled)

Hands-on 1 — Change the window length

Keep the rate fixed and change only the window to see how λ moves.

Q1. Inquiries arrive at an average of 6 per hour. What is λ for a 30-minute window?

Q2. Keeping the same rate of 6 per hour, what is λ for a 2-hour window?

Independent Poissons add

When you want the total count from separate services, or from adjacent intervals rolled together, independent Poissons let you add their mean counts directly.

X ~ Poisson(λ₁), Y ~ Poisson(λ₂) → X + Y ~ Poisson(λ₁ + λ₂)

Thanks to this property, you can roll up fine-grained observations into a coarser bucket, or consolidate several event streams into one.

Hands-on 2 — Add independent Poissons

Independent Poissons can be combined by adding their mean counts.

Q1. Per 10 minutes, the web side fires 1.2 events on average and the DB side fires 0.8 events, independently. What is λ for the total count over the same 10 minutes?

Q2. A system has λ = 0.5 per 5 minutes. Keeping the same rate, what does λ become for a 20-minute window?

"Stretching" and "adding" intervals are the same view

Placing two 30-minute windows side by side to form a 60-minute window is equivalent whether you read it as one long interval or as two short intervals added. Either way, λ adds — and that fact is surprisingly practical in real work.

Hands-on 3 — Combine adjacent intervals

"Stretching" and "adding" intervals are the same view. Either way, λ adds.

Q1. Two consecutive 15-minute windows each have λ = 1.5. What is λ for the combined 30-minute window?

Key takeaways from Chapter 4

  • Change the window length and λ scales proportionally.
  • Independent Poisson distributions are combined by summing their λ values.
  • Granularity changes and summed counts are where Poisson shines in practice.
  • Whenever λ moves, the mean, variance, and relative spread from Chapter 3 move with it.