The Poisson Distribution Topic Review summary

Key learning points

• $P(X=k)=e^{-\\lambda}\\frac{\\lambda^k}{k!}$.
• $E(X)=\\mathrm{Var}(X)=\\lambda$.
• Models rare events in intervals/areas under steady-rate assumptions.
• Sum of independent Poissons is Poisson with summed means.
• Binomial → Poisson when $n$ large, $p$ small, $\\lambda=np$.

Worked example 1 — P(0)

$\\mathrm{Po}(3)$, $P(X=0)$.

Working

First term of the PMF.

Worked example 2 — Mean=Var

Quick Poisson check.

Working

Rough model diagnostic.

Worked example 3 — P(X=k)=e^{-\\\\lambda}\\\\\\\lambda^k{k!}

$P(X=k)=e^{-\\\\lambda}\\\\frac{\\\\lambda^k}{k!}$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — E(X)=\\\Var(X)=\\\\lambda

$E(X)=\\\\mathrm{Var}(X)=\\\\lambda$.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.