| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Moment generating function problems |
| Difficulty | Standard +0.3 This is a standard S4/Further Maths Statistics question requiring routine application of MGF definitions and techniques. Part (i) involves summing a geometric series (straightforward), part (ii) requires understanding convergence conditions (standard knowledge), and part (iii) uses standard differentiation of MGFs. While it's Further Maths content, it follows a well-established template with no novel problem-solving required. |
| Spec | 5.03c Calculate mean/variance: by integration5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(e^{tX}) = \sum_{0}^{\infty} e^{tx}\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)^x\) | M1 | |
| \(= \frac{3}{4}\left(\sum_{0}^{\infty}\left(\frac{e^t}{4}\right)^x\right)\) | M1 | Establish that series is a GP involving \(t\) |
| \(\frac{3}{4}\left(\dfrac{1}{1-\frac{1}{4}e^t}\right)\) | M1 | Use formula for sum to infinity of GP |
| \(\dfrac{3}{4-e^t}\) AG | A1 | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{4}e^t < 1\) | M1 | Allow \( |
| \(t < \ln 4\) | A1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(M'_X(t) = 3e^t(4-e^t)^{-2}\), \(E(X) = 3 \times 1 \times 3^{-2} = \frac{1}{3}\) | M1 | M1 for diffn and sub \(t=0\) |
| A1 | cwo. \((4-e^t)^{-2}\) seen | |
| \(M''_X(t) = \dfrac{3(4-e^t)^2 e^t + 6e^{2t}(4-e^t)}{(4-e^t)^4}\) | M1 | For diffn using prod/quot rules and sub \(t=0\) |
| \(E(X^2) = \dfrac{3 \times 3^2 \times 1 + 6 \times 1 \times 3}{3^4} = \frac{5}{9}\) | M1 | Dep \(+\)ve var |
| \(\text{Var}(X) = \frac{5}{9} - \left(\frac{1}{3}\right)^2 = \frac{4}{9}\) | A1 | |
| [5] |
# Question 5:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(e^{tX}) = \sum_{0}^{\infty} e^{tx}\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)^x$ | M1 | |
| $= \frac{3}{4}\left(\sum_{0}^{\infty}\left(\frac{e^t}{4}\right)^x\right)$ | M1 | Establish that series is a GP involving $t$ |
| $\frac{3}{4}\left(\dfrac{1}{1-\frac{1}{4}e^t}\right)$ | M1 | Use formula for sum to infinity of GP |
| $\dfrac{3}{4-e^t}$ AG | A1 | |
| | **[4]** | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{4}e^t < 1$ | M1 | Allow $|\,|$ for M1, but not A1. $t < 1.39$ |
| $t < \ln 4$ | A1 | |
| | **[2]** | |
## Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $M'_X(t) = 3e^t(4-e^t)^{-2}$, $E(X) = 3 \times 1 \times 3^{-2} = \frac{1}{3}$ | M1 | M1 for diffn and sub $t=0$ |
| | A1 | cwo. $(4-e^t)^{-2}$ seen |
| $M''_X(t) = \dfrac{3(4-e^t)^2 e^t + 6e^{2t}(4-e^t)}{(4-e^t)^4}$ | M1 | For diffn using prod/quot rules and sub $t=0$ |
| $E(X^2) = \dfrac{3 \times 3^2 \times 1 + 6 \times 1 \times 3}{3^4} = \frac{5}{9}$ | M1 | Dep $+$ve var |
| $\text{Var}(X) = \frac{5}{9} - \left(\frac{1}{3}\right)^2 = \frac{4}{9}$ | A1 | |
| | **[5]** | |
---5 The discrete random variable $X$ is such that $\mathrm { P } ( X = x ) = \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { x } , x = 0,1,2 , \ldots$.\\
(i) Show that the moment generating function of $X , \mathrm { M } _ { X } ( t )$, can be written as $\mathrm { M } _ { X } ( t ) = \frac { 3 } { 4 - \mathrm { e } ^ { t } }$.\\
(ii) Find the range of values of $t$ for which the formula for $\mathrm { M } _ { X } ( t )$ in part (i) is valid.\\
(iii) Use $\mathrm { M } _ { X } ( t )$ to find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{OCR S4 2017 Q5 [11]}}