📘 Section 3.8 — Poisson Distribution (PMF • CDF • Excel • Calculator)

Poisson Distribution

A count variable X that records the number of events in a fixed interval with constant average rate λ follows a Poisson(λ) distribution.

PMF (exactly x): P(X = x) = e^−λ · λ^x / x!, x = 0, 1, 2, …

CDF (≤ x): P(X ≤ x) = Σ_k=0^x e^−λ · λ^k / k!

Mean: μ = λ Variance: σ² = λ

🎯 When to use Poisson

Counting events in a fixed interval of time, area, length, or volume.
Events happen independently and randomly (no big batches or cooldown).
The average event rate is constant. Get λ by rate × interval.

Discrete vs Continuous: For Poisson (discrete), P(X=x) is a true probability. For continuous models, the PDF f(x) is a density and probabilities are areas under the curve.

🔁 Rate → λ Converter (how you get λ)

I observe events per and I want the count in a window of λ = –

Example: 1 per 10 minutes in a 60-minute window ⇒ λ = 6. Use the buttons above to try it.

🧮 Interactive Calculator — PMF • CDF • Right tail • Range

λ (rate for the interval) Check probability at k (marks k on charts) μ = 11.500, σ = 3.391

PMF — P(X = x)

CDF — P(X ≤ x)

Point P(X = k): –

Left tail P(X ≤ k): –

Right tail P(X ≥ k): –

Range probability — compute P(a ≤ X ≤ b)

a b –

🔌 / ⚙️ / 🎓 Examples — know when Poisson fits

Electrical / Computer Engineering

Packets per second: count packets in a 1-s window → λ = avg packets/s × 1 s.
Bit errors per frame: if errors are rare and independent over long links.
Decay counts: particle hits per minute at a constant rate.

Mechanical / Manufacturing

Flaws along wire: flaws per mm × length → λ. Count flaws in the piece.
Surface pits per part: random defects over area (use area × rate).

Student life

Emails per hour during a steady period.
Typos per page when the typo rate is roughly constant.

🔧 Worked example — Wire flaws

Flaws occur at 2.3 per mm. For a 5 mm piece, λ = 2.3 × 5 = 11.5.

Exactly 10 flaws: P(X=10) = e^−11.5 · 11.5¹⁰ / 10! = …
Excel (modern): =POISSON.DIST(10, 11.5, FALSE) Legacy: =POISSON(10, 11.5, FALSE)
At least one flaw in 2 mm: λ = 2.3 × 2 = 4.6 → P(X ≥ 1) = 1 − e^−4.6 = …
Excel (modern): =1-POISSON.DIST(0, 4.6, TRUE) Legacy: =1-POISSON(0, 4.6, TRUE)

🏥 Practice — ER arrivals

Average 3 patients/hour → λ = 3 per hour.

Exactly 5 in an hour: P(X=5) = e⁻³ · 3⁵ / 5! = …
Excel (modern): =POISSON.DIST(5, 3, FALSE) Legacy: =POISSON(5, 3, FALSE)
At most 2: P(X ≤ 2) = …
Excel (modern): =POISSON.DIST(2, 3, TRUE) Legacy: =POISSON(2, 3, TRUE)

🧠 ER arrivals — Step-by-step Poisson math

Let X be the number of arrivals in one hour. If arrivals are independent and the average rate is constant, then X ~ Poisson(λ). Here the model uses λ arrivals per hour.

λ (per hour) Mean = λ, Variance = λ

Exactly 5 in an hour:

P(X = 5) = e^{-λ} · λ^5 / 5!
         = [e^{-λ} = …] × [λ^5 / 5! = …]
         = …

Interpretation: probability of seeing exactly 5 arrivals in one hour.

At most 2 in an hour:

P(X ≤ 2) = ∑_{k=0}^2 e^{-λ} · λ^k / k!
         = e^{-λ} · (1 + λ + λ^2 / 2)
         = [sum = …] × [e^{-λ} = …]
         = …

Interpretation: probability of seeing 0, 1, or 2 arrivals in one hour.

What does P(·) mean? Discrete vs continuous (quick note)

Discrete: Poisson uses a PMF, so P(X=k) is an actual probability and the bars sum to 1.
Continuous: PDFs are densities; probabilities are areas (integrals). For a single point, P(X=a)=0.

📎 Excel Cheat Sheet (Poisson)

PMF (exactly x): =POISSON.DIST(x, lambda, FALSE)
CDF (≤ x): =POISSON.DIST(x, lambda, TRUE)
Right tail (≥ k): =1 - POISSON.DIST(k-1, lambda, TRUE)
Range a..b: =POISSON.DIST(b,lambda,TRUE) - POISSON.DIST(a-1,lambda,TRUE)
Mean & Variance: both equal lambda (so =SQRT(lambda) for σ)