Notes

M1: For writing or using \(\mathrm{G}_X(1) = 1\)

A1: For finding \(k = \dfrac{2}{3}\) (must be exact)

Notes

M1: Attempt to differentiate \(\mathrm{G}_X(t)\) to obtain \(A\left(1 - \dfrac{2t}{5}\right)^{-2}\) (o.e.) for some constant \(A\)

A1: Correct differential (may be unsimplified)

A1: For finding [\(\mathrm{E}(X) =\)] \(\dfrac{10}{9}\) (must be exact, and must come from a correct differential)

Notes

M1: For \(\mathrm{G}''_X(t)\) in the form \(A\left(1 - \dfrac{2t}{5}\right)^{-3}\) or a correct expression (not value) for \(\mathrm{G}''_X(0)\)
Series method: Two terms correct from: \(\dfrac{1}{3} + (-1)\left(-\dfrac{2}{5}t\right) + \dfrac{(-1)(-2)}{2}\left(-\dfrac{2}{5}t\right)^2\)

M1: For correct method to find \(\mathrm{G}''_X(t)\) and use of \(\mathrm{P}(X = 2) = \dfrac{\mathrm{G}''_X(0)}{2!}\) (ft their \(\mathrm{G}'_X(t)\) in (b) provided \(\mathrm{G}'_X(t)\) not constant)
Series method: Correct unsimplified expression for \(t^2\) seen (may be part of expansion)

A1: For \(\dfrac{4}{25}\) o.e.

Notes

M1: Sight or use of \(\dfrac{1}{t}\mathrm{G}_X(t)\) or \(\mathrm{G}_X(t^4)\) stated or effected in expression

A1ft: Correct expression, ft their \(k\) (accept \(k\) itself). ISW following a correct expression.

Notes

1^st M1 for attempt to differentiate leading to \(k(4 - 3t)^{-1.5}\);   1^st A1 for \(\mathrm{E}(X) = 1.5\) or exact equivalent

2^nd M1 for attempting to differentiate again leading to \(m(4 - 3t)^{-2.5}\)

2^nd A1ft for \(\dfrac{27}{4}\) or correct ft from their \(k\) provided both Ms are scored

3^rd M1 for a correct method for finding \(\mathrm{Var}(X)\); can ft their \(\dfrac{3}{2}\) and their \(\dfrac{27}{4}\)

3^rd A1 for 6

(corrected from the printed mark scheme: the scheme prints “[So \(\mathrm{E}(X)\) =] \(\mathrm{G}'_X(t) = \dfrac{3}{2}\)”; it should be \(\mathrm{G}'_X(1)\))

Notes

1^st M1 for Maclaurin to find \(\mathrm{P}(X = 2)\) condone \(\text{“}\dfrac{27}{4}\text{”} \times \dfrac{1}{32}\left[= \dfrac{27}{128} \text{ or } 0.2109375\right]\)
or putting in the form \(a(1 - 0.75t)^{-0.5}\)

1^st A1 for a correct unsimplified prob for \(\mathrm{P}(X = 2)\) (may be in binomial expansion) Allow 0.105(468…)

2^nd M1 for use of pgf to find \(\mathrm{P}(X = 1)\) and \(\mathrm{P}(X = 0)\) or attempt 1^st 3 terms of bin expansion

2^nd A1 for \(\dfrac{203}{256}\) or exact equivalent.

Notes

1^st M1 for using product of pgf or multiplication by \(t\)

1^st A1 for correct unsimplified form of pgf

2^nd dM1 (dep. on 1^st M1) for attempting to convert pgf to form given in the formula book
or for stating Geometric alongside a correct PGF for \(Y\) (may be unsimplified)

2^nd A1 for correctly deducing the distribution of \(Y\) as geometric with \(p = 0.25\) [may be seen in (d)]
NB: Final A1 dependent on previous marks being awarded.

(corrected from the printed mark scheme: the scheme also prints \(\mathrm{G}_Y(t) = \dfrac{\frac{1}{4}}{1 - \frac{3}{4}t}\) as an alternative form; this is missing the factor \(t\) and is not the pgf of \(Y\), so it has been left out)

Notes

M1 for attempting to use geometric formula or their pgf (using correct coefficients); A1 for awrt 0.178

Notes

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

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

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

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

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

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

Notes

M1 A correct un-simplified pgf based on \(\sum t^v\mathrm{P}(V = v)\)

A1* cso must see an un-simplified version i.e. M1 scored and no incorrect working seen

Notes

(i) M1 Differentiating using the product rule to find \(\mathrm{G}_W{}^{\prime}(t)\)   Allow un-simplified e.g. \(5 \times \dfrac{2}{5}t\)
Need two terms added and at least one correct. If they expand we need 3 correct.

A1 3 from a correct derivative

(ii) 1^st M1 Attempt \(\mathrm{G}_W{}^{\prime\prime}(t)\) ft their \(\mathrm{G}_W{}^{\prime}(t)\) [must be at least 2 terms or a product], one correct ft term, same rule for differentiating a product

1^st A1 \(\dfrac{36}{5}\) or 7.2 from a correct derivative

2^nd M1 \(\mathrm{G}_W{}^{\prime\prime}(1) + \mathrm{G}_W{}^{\prime}(1) - \left(\mathrm{G}_W{}^{\prime}(1)\right)^2\) ft their \(\mathrm{G}_W{}^{\prime\prime}(t)\) if different from \(\mathrm{G}_W{}^{\prime}(t)\) and \(\mathrm{G}_W(t)\)

2^nd A1 Dep on M3A2 \(\dfrac{6}{5}\) or 1.2

Alternative for (b)

SC MR They use \(\mathrm{G}_V(t)\) instead of \(\mathrm{G}_W(t)\) Provided some correct differentiation seen:
Award B1 for \(\mathrm{E}(V) = \dfrac{14}{5}\) and B1 for \(\mathrm{Var}(V) = \dfrac{12}{25}\) score as M0A1M0A0M0A1

Notes

M1 Realising the need to use \(\mathrm{G}_X(t) = \mathrm{G}_V(t) \times \mathrm{G}_W(t)\)

A1 \(t^3\left(\dfrac{2}{5}t + \dfrac{3}{5}\right)^7\)

Notes

M1 Realising the need to multiply through by \(t^3\) or substitute \(t^2\) for \(t\) or sight of \(t^3\mathrm{G}_X(t^2)\)

A1 \(t^9\left(\dfrac{2}{5}t^2 + \dfrac{3}{5}\right)^7\)   oe   Need not be in its simplest form

Notes

M1 Attempting to find correct coefficient of \(t^n\) or identify \(Y = 2J + 9\) where \(J \sim \mathrm{B}(7, 0.4)\)
Need an expression can ft their \(\mathrm{G}_Y(t)\) or \(\mathrm{G}_X(t)\) of the form \(t^n(at^m + b)^k\)
Allow a statement that \(\mathrm{P}(Y = 15) = 0\) if it follows from their pgf

A1 For a correct exact answer or allow awrt 0.2903   Allow 0.29 from correct expression

Notes

M1: Stating \(\mathrm{G}_X(1) = 1\)   eg   \(\mathrm{G}_X(1) = k(1 + 2)^5 = 1 \quad k(1 + 2)^5 = 1\)
Allow Verification \(\dfrac{1}{243} \times 3^5 = 1\)

A1*: Fully correct proof with no errors   Substituting \(t = 1\)
Verification need therefore \(\mathrm{G}_X(1) = 1\)

Notes

M1: Attempting to find the coefficient of \(t^2\)

A1: \(\dfrac{40}{243}\) or awrt 0.165

Notes

M1: Realising the need to multiply through by \(t^3\) or subst \(t^2\) for \(t\)

A1: \(\dfrac{t^3}{243}\left(1 + 2t^2\right)^5\) oe   eg \(\dfrac{t^3}{243}\left(1 + 10t^2 + 40t^4 + 80t^6 + 80t^8 + 32t^{10}\right)\)

Notes

M1: Realising the need to use \(\mathrm{G}_U(t) = \mathrm{G}_X(t) \times \mathrm{G}_Y(t)\)

A1: \(\dfrac{t(1 + 2t)^7}{2187}\) oe

Notes

M1: For an attempt to differentiate G (\(u\)) e.g \(\mathrm{G}_U{}^{\prime}(t) = At(1 + 2t)^6 + B(1 + 2t)^7\) ft their part(d) if in the form \(kt(1 + 2t)^n\) where \(n \geqslant 5\)

A1ft: \(\dfrac{17}{3}\) or awrt 5.67

M1: For attempting second derivative eg \(\mathrm{G}_U{}^{\prime\prime}(t) = Ct(1 + 2t)^5 + D(1 + 2t)^6\) ft their part(d) if in the form \(kt(1 + 2t)^n\) where \(n \geqslant 5\)

A1 28

M1: Using \(\mathrm{G}_U{}^{\prime\prime}(1) + \mathrm{G}_U{}^{\prime}(1) - \left(\mathrm{G}_U{}^{\prime}(1)\right)^2\) ft their values

A1: \(\dfrac{14}{9}\) or awrt 1.56

Alternative (e)

Notes

B1: 0 (Since there is no term in \(t^3\))

Notes

(i) M1: Realising that \(\tfrac{1}{64}b^2\), the coefficient of \(t^4\), is needed

M1: Equating their coefficient of \(t^4\) to \(\tfrac{25}{64}\) with an attempt to find \(b\)

A1: \(b = 5\) only

M1: Realising that \(\mathrm{G}_X(1) = 1\) is required

A1: \(a = 3\) only

M1: Finding coefficient of \(t^2\) with their \(a \gt 0\) and \(b \gt 0\)

A1: \(\tfrac{15}{32}\)   (condone awrt 0.469)

(ii) M1: Realising \(\mathrm{G}'_X(1)\) is needed

M1: Attempt to differentiate \(\mathrm{G}_X(t)\) with their values of \(a\) and \(b\)

A1ft: 2.5 (ft (3sf) their values of \(a\) and \(b\), \(a \gt 0\) and \(b \gt 0\))   \(\mathrm{E}(X) = \tfrac{ab + b^2}{16}\)

Alternative:
M1: Realising \(X = 0\), 2 and 4 only
M1: \([0 \times \mathrm{P}(X = 0)] + 2 \times \mathrm{P}(X = 2) + 4 \times \mathrm{P}(X = 4)\)

Notes

M1: either \(\mathrm{G}_X(t^3)\) or \(\times t^2\) or using \(Y = 2, 8, 14\)

A1ft: ft their values of \(a\) and \(b\), \(a \gt 0\) and \(b \gt 0\)
\(\mathrm{G}_Y(t) = \tfrac{t^2}{64}(\text{“}3\text{”} + \text{“}5\text{”}t^6)^2\) or \(\mathrm{G}_Y(t) = \tfrac{t^2}{64}(\text{“}9\text{”} + \text{“}30\text{”}t^6 + \text{“}25\text{”}t^{12})\) or
\(\mathrm{G}_Y(t) = \tfrac{1}{64}(\text{“}9t^2\text{”} + \text{“}30\text{”}t^8 + \text{“}25\text{”}t^{14})\)

Notes

B1 for finding \(k\)  (must be exact)

Notes

1^st M1 for an attempt to differentiate \(\mathrm{G}(t)\) e.g. \(A(2 - t)^{-1}\) (o.e.)

1^st A1 for a correct first derivative  (condone \(k\) or use of \(\frac{1}{\ln 2} =\) awrt 1.44)

2^nd A1 for correct \(\mathrm{E}(X)\) or \(\mathrm{G}^{\prime}(1)\)  (allow awrt 1.44    calc: \(1.442695\ldots\) but not \(k\)) seen anywhere

2^nd M1 for attempting second derivative (ft their \(\mathrm{G}^{\prime}(t)\))

3^rd A1 for a correct 2^nd derivative  (condone \(k\) or use of \(\frac{1}{\ln 2} =\) awrt 1.44)

3^rd M1 for a correct method for \(\mathrm{Var}(X)\) (some substitution into the correct formula)

4^th A1 for \(\dfrac{1}{\ln 2}\left(2 - \dfrac{1}{\ln 2}\right)\) o.e. but must simplify i.e. collect like terms
[Mark final answer – penalise incorrect log work etc]
NB  \(0.8040211\ldots\) is A0 unless exact answer seen

Notes

1^st M1 for a suitable strategy to solve the problem (finding link with Maclaurin)
Need mention of coefficient of \(t^3\) and [\(\mathrm{G}^{\prime\prime\prime}(t)\) or \(\mathrm{G}^{\prime\prime\prime}(0)\)] (condone \(\mathrm{G}^{\prime\prime\prime}(1)\))

1^st A1ft for 3^rd derivative, ft their 2^nd derivative in (b) (provided \(\mathrm{G}^{\prime\prime}(t)\) not const)
Correct \(\mathrm{G}^{\prime\prime\prime}(t)\) or \(\mathrm{G}^{\prime\prime\prime}(0)\) scores 1^st M1 1^st A1ft

2^nd M1 for translating Maclaurin to probability (a correct expression)

2^nd A1 for \(\frac{1}{24 \ln 2}\) or awrt 0.0601

ALT (c)  Log series

1^st M1 attempt to write \(\mathrm{G}(t)\) in suitable form as far as: \(k\left[\ln 2 - \ln\left(2\left[1 - \tfrac{t}{2}\right]\right)\right]\)

1^st A1 reaching \(-k \ln\left(1 - \tfrac{t}{2}\right)\)

2^nd M1 use of \(-\ln(1 - x)\) series (some correct substitution)  NB \(\mathrm{G}(t) = \tfrac{1}{\ln 2}\left(\tfrac{t}{2} + \tfrac{t^2}{8} + \tfrac{t^3}{24} + \ldots\right)\)