Discrete Random Variables summary

Key learning points

• $\sum P(X=x)=1$ with each probability non-negative.
• $E(X)=\sum xP(X=x)$; $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$.
• $E(aX+b)=aE(X)+b$; $\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)$.
• Cumulative distribution $F(x)=P(X\le x)$.

Worked example 1 — Expectation

$P(1)=0.2$, $P(2)=0.5$, $P(3)=0.3$. Find $E(X)$.

Working

Weight each value by its probability.

Worked example 2 — with each probability non-negative

Use this method from the Discrete Random Variables sheet: $\sum P(X=x)=1$ with each probability non-negative.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Key skill 2

Use this method from the Discrete Random Variables sheet: $E(X)=\sum xP(X=x)$; $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Key skill 3

Use this method from the Discrete Random Variables sheet: $E(aX+b)=aE(X)+b$; $\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Cumulative distribution

Use this method from the Discrete Random Variables sheet: Cumulative distribution $F(x)=P(X\le x)$.

Working

Each key learning point matches a question type on the printable worksheet.

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.