Discrete Random Variables Topic Review summary

Key learning points

• $\\sum P(X=x)=1$ over the support.
• $E(X)=\\sum x P(X=x)$; $E(g(X))=\\sum g(x)P(X=x)$.
• $\\mathrm{Var}(X)=E(X^2)-[E(X)]^2$.
• Linear: $E(aX+b)=aE(X)+b$, $\\mathrm{Var}(aX+b)=a^2\\mathrm{Var}(X)$.
• Cumulative $F(x)=P(X\\le x)$ steps at mass points.

Worked example 1 — Expectation

$X$ takes 1,2 with probs 0.4,0.6. $E(X)$?

Working

$\\sum x p$.

Worked example 2 — Variance

Use $E(X^2)-\\mu^2$.

Working

Standard exam route.

Worked example 3 — \\\\sum P(X=x)=1 over the support

$\\\\sum P(X=x)=1$ over the support.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — E(X)=\\\\sum x P(X=x); E(g(X))=\\\\sum g(x

$E(X)=\\\\sum x P(X=x)$; $E(g(X))=\\\\sum g(x)P(X=x)$.

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.