A2 June 2023 Q6

EdexcelCurrent spec13 marksGeometric & Negative Binomial Probability Generating Functions

6. The discrete random variable \(X\) has probability generating function

\[\mathrm{G}_X(t) = \frac{t^2}{(3 - 2t)^2}\]

(a) Specify the distribution of \(X\) (2)

A fair die is rolled repeatedly.

(b) Describe an outcome that could be modelled by the random variable \(X\) (1)

(i) \(\mathrm{E}(X)\)

(ii) \(\mathrm{Var}(X)\)

(7)

The discrete random variable \(Y\) has probability generating function

\[\mathrm{G}_Y(t) = \frac{t^{10}}{(3 - 2t^3)^2}\]

(d) Find the exact value of \(\mathrm{P}(Y = 19)\) (3)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{NegBin}(r, p)\) has pgf \(\left[\dfrac{pt}{1 - (1 - p)t}\right]^r\) and identify the connection	M1	2.1
NegBin\(\left(2, \tfrac{1}{3}\right)\)	A1	2.2a
	(2)

Notes

M1 for identifying the NegBin distribution (allow NB for NegBin)

A1 for \(r = 2\) and \(p = \tfrac{1}{3}\)

Mark scheme (b)

Scheme	Marks	AO
e.g. no. of rolls to achieve 5 or 6 (so that \(p = \tfrac{1}{3}\)) twice (oe)	B1ft	3.3
	(1)

Notes

B1ft for identifying a suitable definition for \(X\) using a (fair) die, with \(p = \tfrac{1}{3}\) and the second occurrence of the event, only ft their NegBin distribution in (a). A finite number of rolls is B0

Mark scheme (c)

Scheme	Marks	AO
(i) \(\mathrm{G}'_X(t) = \dfrac{2t(3 - 2t)^2 - (-2) \times 2(3 - 2t)t^2}{(3 - 2t)^4}\) or \(\dfrac{6t}{(3 - 2t)^3}\)	M1 A1	2.1 1.1b
\(\underline{\mathrm{E}(X)} = \mathrm{G}'_X(1) = \underline{\mathbf{6}}\)	A1	1.1b
(ii) \(\mathrm{G}''_X(t) = \dfrac{6(3 - 2t)^3 - (-2) \times 3(3 - 2t)^2 \times 6t}{(3 - 2t)^6}\) or \(\dfrac{18 + 24t}{(3 - 2t)^4}\)	M1	2.1
\(\mathrm{G}''_X(1) = 42\)	A1	1.1b
\(\mathrm{Var}(X) = \text{“}42\text{”} + \text{“}6\text{”} - \text{“}6\text{”}^2\)	M1	1.1b
\(= \underline{\mathbf{12}}\)	A1	1.1b
	(7)

Notes

(i) 1^st M1 for attempt to differentiate quotient or product. At least one \(uv'\) style term correct.

1^st A1 for a fully correct first derivative (needn’t be simplified)

2^nd A1 for \(\mathrm{E}(X) = 6\) NB this A1 depends on M1 only but M1A0A1 is possible

(ii) 2^nd M1 for attempt to diff’ quotient or product again. At least one \(uv'\) style term correct.

3^rd A1 for 42 (may be given for incorrect \(\mathrm{G}''\) provided their \(\mathrm{G}''(1)\) gives 42 and M1 scored)
Note all powers of \((3 - 2t)\) equal 1 when \(t = 1\) is substituted so can be used as a check

3rd M1 for correct use of pgf to find \(\mathrm{Var}(X)\)

4^th A1 dep on M3 for 12

Mark scheme (d)

Scheme	Marks	AO
\(\mathrm{G}_Y(t) = t^{10} \times \dfrac{1}{9}\left[1 - \dfrac{2}{3}t^3\right]^{-2} = \dfrac{t^{10}}{9}\left[1 + \ldots \dfrac{(-2)(-3)(-4)}{3!}\left(-\dfrac{2}{3}\right)^3 t^9 \ldots\right]\)	M1 A1	2.1 1.1b
\(\mathrm{P}(Y = 19) = \dfrac{32}{243}\)	A1	1.1b
	(3)
	(13 marks)

Notes

M1 for writing pgf in suitable form to carry out binomial expansion

1^st A1 for a correct expression for coefficient of \(t^{19}\)

2^nd A1 for \(\frac{32}{243}\) or exact equivalent

Alternative

Scheme	Marks
Identify that \(Y = 3X + 4\)	M1
(\(Y = 19\) requires \(X = 5\) so) \(\mathrm{P}(X = 5) = \dbinom{4}{1}\left(\dfrac{1}{3}\right)\left(\dfrac{2}{3}\right)^3\left(\dfrac{1}{3}\right)\)	A1

ALT M1 for identifying connection \(Y = 3X + 4\)
1^st A1 for a correct numerical probability expression for \(\mathrm{P}(X = 5)\)