| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Determine constant in PGF |
| Difficulty | Challenging +1.2 This is a Further Maths S4 question on PGFs requiring multiple standard techniques: using the PGF property that G(1)=1 to find a relationship, differentiation for expectation, series expansion, and distribution recognition. While it involves several parts and Further Maths content (inherently harder), each step follows routine procedures without requiring novel insight—the geometric series expansion and geometric distribution recognition are standard for this topic. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a - b = 1\) oe, isw this part | B1 | Allow \(\frac{1}{a-b} = 1\). Use \(G_X(1) = 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{(a - bt + bt)}{(a-bt)^2}\) | M1 | Use quotient or product rule. NOT with \(\frac{t}{a-at}\) |
| Use \(G'_X(1) = 1\) | M1 | |
| \(a\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Binomial expansion | M1 | Ignore errors in setting up \((1+\ldots)^{-1}\) for M1. Or \(t(1-b+b^2)+t^2(b-2b^2)+b^2t^3\) |
| \(\dfrac{t}{a} + \dfrac{bt^2}{a^2} + \dfrac{b^2t^3}{a^3}\) | A2 | A1 for 2 terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Geo}\!\left(\tfrac{1}{a}\right)\) | B1, B1ft | \(\frac{1}{a}\) ft B1 |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a - b = 1$ oe, isw this part | B1 | Allow $\frac{1}{a-b} = 1$. Use $G_X(1) = 1$ |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{(a - bt + bt)}{(a-bt)^2}$ | M1 | Use quotient or product rule. NOT with $\frac{t}{a-at}$ |
| Use $G'_X(1) = 1$ | M1 | |
| $a$ | A1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Binomial expansion | M1 | Ignore errors in setting up $(1+\ldots)^{-1}$ for M1. Or $t(1-b+b^2)+t^2(b-2b^2)+b^2t^3$ |
| $\dfrac{t}{a} + \dfrac{bt^2}{a^2} + \dfrac{b^2t^3}{a^3}$ | A2 | A1 for 2 terms correct |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Geo}\!\left(\tfrac{1}{a}\right)$ | B1, B1ft | $\frac{1}{a}$ ft B1 |
---3 The discrete random variable $X$ has probability generating function $\frac { t } { a - b t }$, where $a$ and $b$ are constants.\\
(i) Find a relationship between $a$ and $b$.\\
(ii) Use the probability generating function to find $\mathrm { E } ( X )$ in terms of $a$, giving your answer as simply as possible.\\
(iii) Expand the probability generating function as a power series, as far as the term in $t ^ { 3 }$, giving the coefficients in terms of $a$ and $b$.\\
(iv) Name the distribution for which $\frac { t } { a - b t }$ is the probability generating function, and state its parameter(s) in terms of $a$.
\hfill \mbox{\textit{OCR S4 2014 Q3 [9]}}