Exam-Style Problems

Mean (E(X)) and variance (Var(X)) using the probability generating function — problem 1 · Maths4U Original · 6 marks

17129

Notes & worked examples

Info

Maths4U Original · M4U-EXT-9231-FPS-T448-P01

X has PGF \(\displaystyle G_{X}(t)= \frac{6+ 7t+ 9t^{2}}{22}\).

Find \(\displaystyle P(X=2)\). Explain your reasoning. [2]
Find E(X). Explain your reasoning. [2]
Find Var(X). Explain your reasoning. [2]

Solution

Hint

X takes nonnegative integer values for an ordinary PGF. Read the coefficient of \(\displaystyle t^{2}\).

Worked solution

(a)

\(\displaystyle [t^2]G_X(t)=\frac{9}{22}\)

Answer: \(\displaystyle \frac{9}{22}\).

Answer: \(\displaystyle \frac{9}{22}\)

(b)

\(\displaystyle G_X'(1)=\frac{25}{22}\)

Answer: \(\displaystyle \frac{25}{22}\).

Answer: \(\displaystyle \frac{25}{22}\)

(c)

\(\displaystyle G_X''(1)+G_X'(1)-[G_X'(1)]^2=\frac{321}{484}\)

Answer: \(\displaystyle \frac{321}{484}\).

Answer: \(\displaystyle \frac{321}{484}\)

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.

Mean (E(X)) and variance (Var(X)) using the probability generating function — problem 2 · Maths4U Original · 6 marks

17130

Notes & worked examples

Info

Maths4U Original · M4U-EXT-9231-FPS-T448-P02

X has PGF \(\displaystyle G_{X}(t)= \frac{7+ 8t+ 10t^{2}}{25}\).

Find \(\displaystyle P(X=2)\). Explain your reasoning. [2]
Find E(X). Explain your reasoning. [2]
Find Var(X). Explain your reasoning. [2]

Solution

Hint

X takes nonnegative integer values for an ordinary PGF. Read the coefficient of \(\displaystyle t^{2}\).

Worked solution

(a)

\(\displaystyle [t^2]G_X(t)=\frac{2}{5}\)

Answer: \(\displaystyle \frac{2}{5}\).

Answer: \(\displaystyle \frac{2}{5}\)

(b)

\(\displaystyle G_X'(1)=\frac{28}{25}\)

Answer: \(\displaystyle \frac{28}{25}\).

Answer: \(\displaystyle \frac{28}{25}\)

(c)

\(\displaystyle G_X''(1)+G_X'(1)-[G_X'(1)]^2=\frac{416}{625}\)

Answer: \(\displaystyle \frac{416}{625}\).

Answer: \(\displaystyle \frac{416}{625}\)

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.