| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2008 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Determine constant in PGF |
| Difficulty | Challenging +1.2 This is a Further Maths S4 question on PGFs requiring multiple standard techniques: using G(1)=1 to find a constant, coefficient extraction for probabilities, using H(t)=[G(t)]³ for sums, and the even/odd probability trick with H(1)±H(-1). While PGFs are an advanced topic, each part follows textbook methods without requiring novel insight—harder than typical A-level due to the Further Maths content but routine for students at this level. |
| Spec | 5.02a Discrete probability distributions: general |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(A = 42\) | B1 | |
| State \(k = \frac{1}{8}\) | B1 | or 0.11 or greater accuracy |
| Attempt correct process for finding m | M1 | involving logarithms or equiv |
| Obtain \(\frac{1}{3}\ln 2\) or 0.077 | A1 | or 0.08 or greater accuracy |
| (ii) Attempt solution for t using either formula | M1 | using correct process (log'ms or T&L or …) |
| Obtain 11.3 | A1 | or greater accuracy; allow 11.3 ± 0.1 |
| (iii) Differentiate to obtain form \(Be^{mt}\) | M1 | where B is different from A |
| Obtain 3.235e^{0.077t} | A1∇ | or equiv; following their A and m |
| Obtain 47.9 | A1 | allow 48 or greater accuracy |
(i) State $A = 42$ | B1 |
State $k = \frac{1}{8}$ | B1 | or 0.11 or greater accuracy
Attempt correct process for finding m | M1 | involving logarithms or equiv
Obtain $\frac{1}{3}\ln 2$ or 0.077 | A1 | or 0.08 or greater accuracy
(ii) Attempt solution for t using either formula | M1 | using correct process (log'ms or T&L or …)
Obtain 11.3 | A1 | or greater accuracy; allow 11.3 ± 0.1
(iii) Differentiate to obtain form $Be^{mt}$ | M1 | where B is different from A
Obtain 3.235e^{0.077t} | A1∇ | or equiv; following their A and m
Obtain 47.9 | A1 | allow 48 or greater accuracy7 The probability generating function of the random variable $X$ is given by
$$\mathrm { G } ( t ) = \frac { 1 + a t } { 4 - t }$$
where $a$ is a constant.\\
(i) Find the value of $a$.\\
(ii) Find $\mathrm { P } ( X = 3 )$.
The sum of 3 independent observations of $X$ is denoted by $Y$. The probability generating function of $Y$ is denoted by $\mathrm { H } ( t )$.\\
(iii) Use $\mathrm { H } ( t )$ to find $\mathrm { E } ( Y )$.\\
(iv) By considering $\mathrm { H } ( - 1 ) + \mathrm { H } ( 1 )$, show that $\mathrm { P } ( Y$ is an even number $) = \frac { 62 } { 125 }$.
\hfill \mbox{\textit{OCR S4 2008 Q7 [13]}}