| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Derive standard distribution PGF |
| Difficulty | Standard +0.8 This is a Further Maths S4 question requiring derivation of the binomial PGF from first principles using the definition G(t) = E(t^X), then applying the independence property of PGFs. Part (i) requires algebraic manipulation with binomial theorem, part (ii) is more routine application. Moderately challenging due to the formal derivation requirement and Further Maths context, but follows standard PGF theory without novel insight. |
| Spec | 5.02a Discrete probability distributions: general5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sum_{x=0}^{n} \binom{n}{x} p^x q^{n-x} t^x\) | M1 | From \(E(t^x)\) |
| \(= \sum_{x=0}^{n} \binom{n}{x} (pt)^x q^{n-x}\) | A1 | M1A0: \(\sum\) without limits |
| 2 | \(G_X(t) = q + pt\); M1 then argument A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(G_T(t) = (q+pt)^n(q+pt)^{2n}\) | M1 | Multiplying pgfs |
| \(= (q+pt)^{3n}\) | A1 | |
| So \(T \sim B(3n, p)\) | M1 | For B |
| A1 4 | For parameters | |
| [6] |
## Question 1:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum_{x=0}^{n} \binom{n}{x} p^x q^{n-x} t^x$ | M1 | From $E(t^x)$ |
| $= \sum_{x=0}^{n} \binom{n}{x} (pt)^x q^{n-x}$ | A1 | M1A0: $\sum$ without limits |
| | **2** | $G_X(t) = q + pt$; M1 then argument A0 |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $G_T(t) = (q+pt)^n(q+pt)^{2n}$ | M1 | Multiplying pgfs |
| $= (q+pt)^{3n}$ | A1 | |
| So $T \sim B(3n, p)$ | M1 | For B |
| | A1 **4** | For parameters |
| | **[6]** | |
---1 The random variable $X$ has the distribution $\mathrm { B } ( n , p )$.\\
(i) Show, from the definition, that the probability generating function of $X$ is $( q + p t ) ^ { n }$, where $q = 1 - p$.\\
(ii) The independent random variable $Y$ has the distribution $\mathrm { B } ( 2 n , p )$ and $T = X + Y$. Use probability generating functions to determine the distribution of $T$, giving its parameters.
\hfill \mbox{\textit{OCR S4 2011 Q1 [6]}}