| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2008 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Moment generating function problems |
| Difficulty | Standard +0.8 This S4 question requires understanding MGF structure to extract probabilities, then using E(X) to solve simultaneous equations, computing variance via E(X²), and interpreting the MGF form. It combines multiple techniques (differentiation, algebraic manipulation, variance formula) with conceptual understanding of MGFs, making it moderately challenging but still within standard Further Maths scope. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Obtain expression of form \(\frac{a \tan \alpha}{b + c \tan^2 \alpha}\) | M1 | any non-zero constants a, b, c |
| State correct \(\frac{2 \tan \alpha}{1 - \tan^2 \alpha}\) | A1 | or equiv |
| Attempt to produce polynomial equation in \(\tan \alpha\) | M1 | using sound process |
| Obtain at least one correct value of \(\tan \alpha\) | A1 | \(\tan \alpha = \pm \frac{1}{\sqrt{3}}\) |
| Obtain 41.8 | A1 | allow 42 or greater accuracy; allow 0.73 allow 138 or greater accuracy |
| Obtain 138.2 and no other values between 0 and 180 | A1 | [SC: Answers only 41.8 or … B1; 138.2 or … no others B1] |
| (b)(i) State \(\frac{2}{6}\) | B1 | |
| (ii) Attempt use of identity linking \(\cot^2 \beta\) and \(\cosec^2 \beta\) | M1 | or equiv retaining exactness; condone sign errors |
| Obtain \(\frac{13}{36}\) | A1 | or exact equiv |
(a) Obtain expression of form $\frac{a \tan \alpha}{b + c \tan^2 \alpha}$ | M1 | any non-zero constants a, b, c
State correct $\frac{2 \tan \alpha}{1 - \tan^2 \alpha}$ | A1 | or equiv
Attempt to produce polynomial equation in $\tan \alpha$ | M1 | using sound process
Obtain at least one correct value of $\tan \alpha$ | A1 | $\tan \alpha = \pm \frac{1}{\sqrt{3}}$
Obtain 41.8 | A1 | allow 42 or greater accuracy; allow 0.73 allow 138 or greater accuracy
Obtain 138.2 and no other values between 0 and 180 | A1 | [SC: Answers only 41.8 or … B1; 138.2 or … no others B1]
(b)(i) State $\frac{2}{6}$ | B1 |
(ii) Attempt use of identity linking $\cot^2 \beta$ and $\cosec^2 \beta$ | M1 | or equiv retaining exactness; condone sign errors
Obtain $\frac{13}{36}$ | A1 | or exact equiv5 The discrete random variable $X$ has moment generating function $\frac { 1 } { 4 } \mathrm { e } ^ { 2 t } + a \mathrm { e } ^ { 3 t } + b \mathrm { e } ^ { 4 t }$, where $a$ and $b$ are constants. It is given that $\mathrm { E } ( X ) = 3 \frac { 3 } { 8 }$.\\
(i) Show that $a = \frac { 1 } { 8 }$, and find the value of $b$.\\
(ii) Find $\operatorname { Var } ( X )$.\\
(iii) State the possible values of $X$.
\hfill \mbox{\textit{OCR S4 2008 Q5 [11]}}