Hint
X takes nonnegative integer values for an ordinary PGF. Read the coefficient of \(\displaystyle t^{2}\).
Worked solution
(a)
\(\displaystyle [t^2]G_X(t)=\frac{12}{31}\)
Answer: \(\displaystyle \frac{12}{31}\).
Answer: \(\displaystyle \frac{12}{31}\)
(b)
\(\displaystyle G_X'(1)=\frac{34}{31}\)
Answer: \(\displaystyle \frac{34}{31}\).
Answer: \(\displaystyle \frac{34}{31}\)
(c)
\(\displaystyle G_X''(1)+G_X'(1)-[G_X'(1)]^2=\frac{642}{961}\)
Answer: \(\displaystyle \frac{642}{961}\).
Answer: \(\displaystyle \frac{642}{961}\)
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: Read the coefficient of \(\displaystyle t^{2}\).
- 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 mistake G″(1) for the second raw moment.
(b) · 2 marks
- 1 mark: Differentiate the PGF once and evaluate at 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: Probabilities are coefficients, not PGF values at the outcome.
(c) · 2 marks
- 1 mark: Convert the second factorial moment to a 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 mistake G″(1) for the second raw moment.
Total: 6 marks.