Hint
A single inverse branch is valid only on a monotone support. Differentiate the transformed CDF.
Worked solution
(a)
\(\displaystyle f_{Y}(y)= F_{Y}'(y)= \frac{1}{121}\)
Answer: \(\displaystyle \frac{1}{121}\).
Answer: \(\displaystyle \frac{1}{121}\)
(b)
\(\displaystyle E(Y)= \frac{121}{2}= \frac{121}{2}\)
Answer: \(\displaystyle \frac{121}{2}\).
Answer: \(\displaystyle \frac{121}{2}\)
(c)
\(\displaystyle \operatorname{Var}(Y)= \frac{(121)^{2}}{12}= \frac{14641}{12}\)
Answer: \(\displaystyle \frac{14641}{12}\).
Answer: \(\displaystyle \frac{14641}{12}\)
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: Differentiate the transformed CDF.
- 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 substitute into the old density without a derivative factor.
(b) · 2 marks
- 1 mark: Recognise a uniform variable or integrate y fY(y).
- 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: \(\displaystyle Y=X^{2}\) needs two branches when X can have either sign.
(c) · 2 marks
- 1 mark: Use the width-squared uniform variance.
- 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 substitute into the old density without a derivative factor.
Total: 6 marks.