| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Derive standard distribution PGF |
| Difficulty | Standard +0.8 This is a multi-part Further Maths Statistics question requiring derivation of a geometric distribution PGF from first principles, understanding convergence conditions, differentiation to find expectation, and applying convolution of PGFs to model a negative binomial scenario. While systematic, it requires strong technical facility with infinite series, PGF manipulation, and connecting distributions—significantly above average A-level difficulty but follows standard S4 patterns. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(= pt/1 – qt)\) AG | M1, A1, A1 | or =pt, r=qt |
| [3] | ||
| (ii) For convergence of the infinite series | B1 | Or \(G\) would be \(\leq 0\) (or probs or denom) |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Mean \(= G'(1) = \ldots =1/p\) | M1 A1 | or product rule. CWO |
| [3] | ||
| (iv) \(G_U = 0.2t/(1 – 0.8t) ; G_X = [0.2t(1 – 0.8t)]^4\) | B1 B1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 0.01024 =32/3125\) | M1, M1, A1 | Or 3 in the first 5 (B(5, 0.2) and 1 in 6th |
| [3] |
**(i)** $E(t^p) = pt + qpt^2 + q^2pt^3 + \ldots.$
$= pt(1 + qt + q^2t^2 + \ldots .$
$= pt/1 – qt)$ AG | M1, A1, A1 | or =pt, r=qt | or $(1-qt)^{-1} = 1+qt+\ldots$
| [3] |
**(ii)** For convergence of the infinite series | B1 | Or $G$ would be $\leq 0$ (or probs or denom)
| [1] |
**(iii)** $G'(t) = [p(1 – qt) + pqt]/(1 – qt)^2$
Mean $= G'(1) = \ldots =1/p$ | M1 A1 | or product rule. CWO
| [3] |
**(iv)** $G_U = 0.2t/(1 – 0.8t) ; G_X = [0.2t(1 – 0.8t)]^4$ | B1 B1 |
| [2] |
**(v)** Find the coefficient of $t^6$ in expansion of $G_X$
$= 0.2^4 \times (4 \times 5/2) \times 0.8^2$
$= 0.01024 =32/3125$ | M1, M1, A1 | Or 3 in the first 5 (B(5, 0.2) and 1 in 6th | =0.0512, x0.2
| [3] |4 The random variable $U$ has the distribution $\operatorname { Geo } ( p )$.\\
(i) Show, from the definition, that the probability generating function ( pgf ) of $U$ is given by
$$G _ { U } ( t ) = \frac { p t } { 1 - q t } , \text { for } | t | < \frac { 1 } { q } ,$$
where $q = 1 - p$.\\
(ii) Explain why the condition $| t | < \frac { 1 } { q }$ is necessary.\\
(iii) Use the pgf to obtain $\mathrm { E } ( U )$.
Each packet of Corn Crisp cereal contains a voucher and $20 \%$ of the vouchers have a gold star. When 4 gold stars have been collected a gift can be claimed. Let $X$ denote the number of packets bought by a family up to and including the one from which the $4 ^ { \text {th } }$ gold star is obtained.\\
(iv) Obtain the pgf of $X$.\\
(v) Find $\mathrm { P } ( X = 6 )$.
\hfill \mbox{\textit{OCR S4 2012 Q4 [12]}}