| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Find probabilities from PGF |
| Difficulty | Challenging +1.2 This is a Further Maths S4 question requiring standard PGF techniques: expanding the generating function to find a coefficient for P(Y=3), and differentiating to find E(Y). While the algebraic manipulation is somewhat involved (geometric series expansion and product differentiation), these are routine procedures for students at this level with no novel problem-solving required. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{126}(64t - t^7)(1 + \frac{t}{2} + \frac{t^2}{4} + ...)\) | M1A1 | M1 for attempt at binomial expansion, as far as term in \(t^2\) |
| \(P(Y=3) = \) coefficient of \(t^3\) | M1 | M1 for coefficient of \(t^3\) seen or implied; attempt to find term in \(t^3\) |
| \(= \frac{16}{126} = \frac{8}{63}\) | A1 | Answer, \(16/126\) or equivalent, allow \(0.127\) |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G'_Y(t) = \frac{1}{126}(64 - 7t^6)(1 - \frac{t}{2})^{-1} + \frac{1}{252}(64t - t^7)(1 - \frac{t}{2})^{-2}\) | M1 | Attempt at product or quotient rule |
| A1 | \(P(1) = \frac{64}{126},\ P(2) = \frac{32}{126},\ ...\ P(6) = \frac{2}{126}\); B1 for \(Y = 1, 2, ..., 6\); B1 for all probabilities correct | |
| \(G'_Y(1) = \frac{40}{21}\) | M1A1 | Sub \(t = 1\) for M1, allow \(1.90\); M1 for \(\sum xp\) used; A1 for \(\frac{40}{21}\) |
| [4] |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{126}(64t - t^7)(1 + \frac{t}{2} + \frac{t^2}{4} + ...)$ | M1A1 | M1 for attempt at binomial expansion, as far as term in $t^2$ |
| $P(Y=3) = $ coefficient of $t^3$ | M1 | M1 for coefficient of $t^3$ seen or implied; attempt to find term in $t^3$ |
| $= \frac{16}{126} = \frac{8}{63}$ | A1 | Answer, $16/126$ or equivalent, allow $0.127$ |
| | **[5]** | |
---
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G'_Y(t) = \frac{1}{126}(64 - 7t^6)(1 - \frac{t}{2})^{-1} + \frac{1}{252}(64t - t^7)(1 - \frac{t}{2})^{-2}$ | M1 | Attempt at product or quotient rule |
| | A1 | $P(1) = \frac{64}{126},\ P(2) = \frac{32}{126},\ ...\ P(6) = \frac{2}{126}$; B1 for $Y = 1, 2, ..., 6$; B1 for all probabilities correct |
| $G'_Y(1) = \frac{40}{21}$ | M1A1 | Sub $t = 1$ for M1, allow $1.90$; M1 for $\sum xp$ used; A1 for $\frac{40}{21}$ |
| | **[4]** | |7 The discrete random variable $Y$ has probability generating function $\mathrm { G } _ { Y } ( t ) = \frac { 1 } { 126 } t \left( 64 - t ^ { 6 } \right) \left( 1 - \frac { t } { 2 } \right) ^ { - 1 }$.\\
(i) Find $\mathrm { P } ( Y = 3 )$.\\
(ii) Find $\mathrm { E } ( Y )$.
\hfill \mbox{\textit{OCR S4 2017 Q7 [9]}}