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Monte

Poisson: counts per unit

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  2. Practice it, not yet
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Poisson counts events in a fixed window when they arrive at a steady average rate λ. Mean and variance both equal λ. Scale λ to the window first.

Why it matters

Poisson counts things that arrive at random but at a steady rate: calls, emails, goals, typos.

It tells you how many staff, servers or spare parts keep you from being caught short.

  • StaffingA desk averaging 4 calls an hour gets 8+ in about 5% of hours.
  • TechServers sized for the average fall over on normal busy minutes.
  • SportsGoals per soccer match are close to Poisson.

Tennis example

You make 3 unforced errors a set on average. P(none in a set) = e⁻³ = 5%. Over 3 sets λ = 9, so P(6 or fewer) is about 21%.

Poisson: drag λ

y = 00.7%
y = 13.4%
y = 28.4%
y = 314.0%
y = 417.5%
y = 517.5%
y = 614.6%
y = 710.4%
y = 86.5%
y = 93.6%
y = 101.8%
y = 110.8%
y = 120.3%
y = 130.1%
y = 140.0%

Mean = variance = 5. P(0) = e−λ = 0.0067. P(at least 2) = 0.9596.

Check yourself

  1. A coffee shop gets 30 orders per hour. λ for 10 minutes?

  2. P(no orders in 10 minutes)?

  3. P(2 or more orders in 10 minutes)?

The trap

The mistake people make most on this one.

Forgetting to rescale λ. 3 calls an hour over 2 hours is λ = 6, not 3.

Formulas

5 formulas and a way to remember them.
Poisson
P(Y = y) = e⁻λ λʸ / y!
Mean and variance
E(Y) = V(Y) = λ
Given at least one
P(Y = k | Y ≥ 1) = P(Y = k) / (1 − P(0)), for k ≥ 1
Rescale
λ for a window = rate × window length
Excel
POISSON.DIST(y, λ, FALSE) for exactly y; TRUE for at most y
Remember it
Poisson counts arrivals: help-desk calls, emails an hour, errors per set. Pick the window first, then count.

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