Hint
Integrate separately across every density breakpoint. Add both triangular areas.
Worked solution
(a)
\(\displaystyle k(\frac{8^{2}}{2}+ \frac{8^{2}}{2})= 1\implies k= \frac{1}{64}\)
Answer: \(\displaystyle \frac{1}{64}\).
Answer: \(\displaystyle \frac{1}{64}\)
(b)
\(E(X)=8\) by symmetry about the centre of the support.
Answer: \(8\).
Answer: \(8\)
(c)
\(\displaystyle \int(x-E(X))^2f(x)\,dx=\frac{32}{3}\)
Answer: \(\displaystyle \frac{32}{3}\).
Answer: \(\displaystyle \frac{32}{3}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Add both triangular areas.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not omit the accumulated area from an earlier branch.
(b) · 2 marks
- 1 mark: Use symmetry or evaluate the two first-moment integrals.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A derivative of F gives the density only on the appropriate support.
(c) · 2 marks
- 1 mark: Integrate the centred square across both branches.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not omit the accumulated area from an earlier branch.
Total: 6 marks.