| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Find PGF from probability distribution |
| Difficulty | Standard +0.8 This is a multi-part S4 question requiring knowledge of PGFs, binomial theorem manipulation, expectation/variance from derivatives, distribution recognition, and understanding of independence. Part (iv) requires insight that G_Y(t) ≠ G_U(t²), and part (v) tests conceptual understanding of when PGFs multiply. More demanding than typical S4 questions but follows standard techniques once the binomial pattern is recognized. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G(t) = E(t^X) = \frac{1}{16}(1+4t+6t^2+4t^3+t^4)\) | M1A1 | Correct form; correct coefficients |
| \(= \frac{1}{16}(1+t)^4\) | A1 | allow \(\left(\frac{1}{2}+\frac{1}{2}t\right)^4\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G'(t) = \frac{1}{4}(1+t)^3\) | M1 | or expanded form. No marks from part (iii) |
| \(E(U) = G'(1) = 2\) | A1 | |
| \(G''(t) = \frac{3}{4}(1+t)^2\) | ||
| \(\text{Var}(U) = G''(1) + G'(1) - (G'(1))^2 = 3+2-4 = 1\) | M1, A1 | Finding \(G''\) and formula correct |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(B(4,\frac{1}{2})\) | B1 | Binomial |
| B1 | Parameters | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Y = 0\ 1\ 4\ 9\ 16\) | B1 | Values of \(Y\) |
| \(G_Y(t) = \frac{1}{16} + \frac{1}{4}t + \frac{3}{8}t^4 + \frac{1}{4}t^9 + \frac{1}{16}t^{16}\) | B1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| No, \(U\) and \(Y\) are not independent | B1 | |
| [1] |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G(t) = E(t^X) = \frac{1}{16}(1+4t+6t^2+4t^3+t^4)$ | M1A1 | Correct form; correct coefficients |
| $= \frac{1}{16}(1+t)^4$ | A1 | allow $\left(\frac{1}{2}+\frac{1}{2}t\right)^4$ |
| **[3]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G'(t) = \frac{1}{4}(1+t)^3$ | M1 | or expanded form. No marks from part (iii) |
| $E(U) = G'(1) = 2$ | A1 | |
| $G''(t) = \frac{3}{4}(1+t)^2$ | | |
| $\text{Var}(U) = G''(1) + G'(1) - (G'(1))^2 = 3+2-4 = 1$ | M1, A1 | Finding $G''$ and formula correct |
| **[4]** | | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B(4,\frac{1}{2})$ | B1 | Binomial |
| | B1 | Parameters |
| **[2]** | | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Y = 0\ 1\ 4\ 9\ 16$ | B1 | Values of $Y$ |
| $G_Y(t) = \frac{1}{16} + \frac{1}{4}t + \frac{3}{8}t^4 + \frac{1}{4}t^9 + \frac{1}{16}t^{16}$ | B1 | |
| **[2]** | | |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| No, $U$ and $Y$ are not independent | B1 | |
| **[1]** | | |
---5 The discrete random variable $U$ has probability distribution given by
$$\mathrm { P } ( U = r ) = \begin{cases} \frac { 1 } { 16 } \binom { 4 } { r } & r = 0,1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find and simplify the probability generating function (pgf) of $U$.\\
(ii) Use the pgf to find $\mathrm { E } ( U )$ and $\operatorname { Var } ( U )$.\\
(iii) Identify the distribution of $U$, giving the values of any parameters.\\
(iv) Obtain the pgf of $Y$, where $Y = U ^ { 2 }$.\\
(v) State, giving a reason, whether you can obtain the pgf of $U + Y$ by multiplying the pgf of $U$ by the pgf of $Y$.
\hfill \mbox{\textit{OCR S4 2013 Q5 [12]}}