Hint
Independence is essential for PGF multiplication. Add the means of the independent binomial variables.
Worked solution
(a)
\(mn p=6\)
Answer: \(6\).
Answer: \(6\)
(b)
\(mn p(1-p)=3\)
Answer: \(3\).
Answer: \(3\)
(c)
\(\displaystyle G_S(0)=(1-p)^{mn}=\frac{1}{4096}\)
Answer: \(\displaystyle \frac{1}{4096}\).
Answer: \(\displaystyle \frac{1}{4096}\)
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 the means of the independent binomial variables.
- 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 add PGFs when adding independent variables.
(b) · 2 marks
- 1 mark: Add their independent variances.
- 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: Different copies of X are not the same variable multiplied by their count.
(c) · 2 marks
- 1 mark: Read the constant coefficient of the product PGF.
- 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 add PGFs when adding independent variables.
Total: 6 marks.