Continuous Random Variables Topic Review summary

Key learning points

• $\\int_{-\\infty}^{\\infty} f(x)\\,dx=1$ with $f(x)\\ge0$.
• $P(a\\le X\\le b)=\\int_a^b f(x)\\,dx$; points have probability 0.
• $F'(x)=f(x)$ when $F$ is the CDF.
• $E(X)=\\int x f(x)\\,dx$; $\\mathrm{Var}(X)=E(X^2)-[E(X)]^2$.
• Medians solve $F(m)=\\frac12$; modes maximise $f$.

Worked example 1 — Normalise

Find $k$ if $f(x)=kx$ on $[0,2]$.

Working

$\\int_0^2 kx\\,dx=1$.

Worked example 2 — CDF

Relation of $F$ and $f$.

Working

Fundamental theorem link.

Worked example 3 — \\\\int_{-\\\\infty}^{\\\\infty} f(x)\\\\,

$\\\\int_{-\\\\infty}^{\\\\infty} f(x)\\\\,dx=1$ with $f(x)\\\\ge0$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — P(a\\\\le X\\\\le b)=\\\\int_a^b f(x)\\\\,

$P(a\\\\le X\\\\le b)=\\\\int_a^b f(x)\\\\,dx$; points have probability 0.

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.