| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Solve for parameters using PGF coefficients |
| Difficulty | Challenging +1.2 This is a multi-part PGF question requiring knowledge of the relationship between PGF derivatives and moments, solving simultaneous equations from probability constraints, and applying convolution properties. While it involves several steps and Further Maths content (S4), the techniques are standard applications of PGF theory without requiring novel insight or complex manipulation. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \(\frac{1}{3}e^{3x} + e^x\) | B1 | |
| Substitute to obtain \(\frac{1}{3}e^{3u} + e^{3u} - \frac{1}{3}e^{-3u} - e^{-3u}\) | B1 | or equiv |
| Equate definite integral to 100 and attempt rearrangement | M1 | as far as \(e^{3u} = \ldots\) |
| Introduce natural logarithm | M1 | using correct process |
| Obtain \(a = \frac{1}{3}\ln(300 + 3e^u - 2e^{3u})\) | A1 5 | AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain correct first iterate | B1 | allow 4 dp rounded or truncated |
| Show correct iteration process | M1 | with at least one more step |
| Obtain at least three correct iterates in all | A1 | allowing recovery after error |
| Obtain 0.6309 | A1 4 | following at least three correct steps; answer required to exactly 4 dp. |
### (i)
Obtain $\frac{1}{3}e^{3x} + e^x$ | B1 |
Substitute to obtain $\frac{1}{3}e^{3u} + e^{3u} - \frac{1}{3}e^{-3u} - e^{-3u}$ | B1 | or equiv |
Equate definite integral to 100 and attempt rearrangement | M1 | as far as $e^{3u} = \ldots$ |
Introduce natural logarithm | M1 | using correct process |
Obtain $a = \frac{1}{3}\ln(300 + 3e^u - 2e^{3u})$ | A1 5 | AG; necessary detail needed |
### (ii)
Obtain correct first iterate | B1 | allow 4 dp rounded or truncated |
Show correct iteration process | M1 | with at least one more step |
Obtain at least three correct iterates in all | A1 | allowing recovery after error |
Obtain 0.6309 | A1 4 | following at least three correct steps; answer required to exactly 4 dp. |
[0.6 → 0.631269 → 0.630884 → 0.630889]4 The probability generating function of the discrete random variable $Y$ is given by
$$\mathrm { G } _ { Y } ( t ) = \frac { a + b t ^ { 3 } } { t }$$
where $a$ and $b$ are constants.\\
(i) Given that $\mathrm { E } ( Y ) = - 0.7$, find the values of $a$ and $b$.\\
(ii) Find $\operatorname { Var } ( Y )$.\\
(iii) Find the probability that the sum of 10 random observations of $Y$ is - 7 .
\hfill \mbox{\textit{OCR S4 2009 Q4 [10]}}