Hint
List the complete support before normalising. Set the sum of all probabilities to one.
Worked solution
(a)
\(\displaystyle 31k=1\implies k=\frac{1}{31}\)
Answer: \(\displaystyle \frac{1}{31}\).
Answer: \(\displaystyle \frac{1}{31}\)
(b)
\(\displaystyle \sum xp(x)=\frac{306}{31}\)
Answer: \(\displaystyle \frac{306}{31}\).
Answer: \(\displaystyle \frac{306}{31}\)
(c)
\(\displaystyle E(X^2)-E(X)^2=\frac{4698}{31}-(\frac{306}{31})^2=\frac{52002}{961}\)
Answer: \(\displaystyle \frac{52002}{961}\).
Answer: \(\displaystyle \frac{52002}{961}\)
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: Set the sum of all probabilities to one.
- 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 negative or zero support values.
(b) · 2 marks
- 1 mark: Weight each support value by its probability.
- 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: Variance is not \(\displaystyle E(X^{2})-E(X)\).
(c) · 2 marks
- 1 mark: Subtract the squared first moment from the second moment.
- 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 negative or zero support values.
Total: 6 marks.