| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moment generating functions |
| Type | Find parameter from PDF |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on standard statistical theory. Parts (i)-(iii) involve routine integration and application of definitions (finding normalizing constant, expectation, unbiasedness, variance of linear combinations). Part (iv) requires comparing variances of two estimators—a standard textbook exercise. While it's Further Maths content, the techniques are mechanical with no novel problem-solving required. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^\infty k(x+\theta)^{-5}\,dx = 1\) | M1 | |
| \(k = 4\theta^4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^\infty 4\theta^4 x(x+\theta)^{-5}\,dx\) | M1 | ft \(k\) |
| Attempt integration by parts or substitution | M1 | |
| \(= \dfrac{\theta}{3}\) AG | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{3}{n}\sum_{1}^{n} X_i = 3E(X) = 3(\frac{\theta}{3}) = \theta\) | B1 | or \(\frac{3}{n} \times n \times \frac{\theta}{3}\) |
| \(\text{Var}(T_1) = 9\Sigma\text{Var}(X)/n^2\) | M1 | or \(\frac{9}{n^2} \times n \times \frac{2\theta^2}{9}\) |
| \(= \dfrac{2\theta^2}{n}\) | A1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var}(T_2) = \frac{1}{9}\text{Var}(X_1) + \text{Var}(X_2) + \frac{25}{9}\text{Var}(X_3)\) | M1 | Allow \(\text{Var}(T_1) = 3\sigma^2\) and \(\text{Var}(T_2) = \dfrac{35\sigma^2}{9}\) |
| \(= \dfrac{70\theta^2}{81}\) | A1 | |
| \(> \dfrac{2\theta^2}{3}\) | M1 | ft their Vars. |
| \(T_1\) is more efficient | A1 | |
| [4] |
# Question 7:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^\infty k(x+\theta)^{-5}\,dx = 1$ | M1 | |
| $k = 4\theta^4$ | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^\infty 4\theta^4 x(x+\theta)^{-5}\,dx$ | M1 | ft $k$ |
| Attempt integration by parts or substitution | M1 | |
| $= \dfrac{\theta}{3}$ AG | A1 | |
## Question (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{3}{n}\sum_{1}^{n} X_i = 3E(X) = 3(\frac{\theta}{3}) = \theta$ | B1 | or $\frac{3}{n} \times n \times \frac{\theta}{3}$ |
| $\text{Var}(T_1) = 9\Sigma\text{Var}(X)/n^2$ | M1 | or $\frac{9}{n^2} \times n \times \frac{2\theta^2}{9}$ |
| $= \dfrac{2\theta^2}{n}$ | A1 | |
| | **[3]** | |
---
## Question (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var}(T_2) = \frac{1}{9}\text{Var}(X_1) + \text{Var}(X_2) + \frac{25}{9}\text{Var}(X_3)$ | M1 | Allow $\text{Var}(T_1) = 3\sigma^2$ and $\text{Var}(T_2) = \dfrac{35\sigma^2}{9}$ |
| $= \dfrac{70\theta^2}{81}$ | A1 | |
| $> \dfrac{2\theta^2}{3}$ | M1 | ft their Vars. |
| $T_1$ is more efficient | A1 | |
| | **[4]** | |7 The continuous random variable $X$ has probability density function
$$f ( x ) = \left\{ \begin{array} { c l }
\frac { k } { ( x + \theta ) ^ { 5 } } & \text { for } x \geqslant 0 \\
0 & \text { otherwise }
\end{array} \right.$$
where $k$ is a positive constant and $\theta$ is a parameter taking positive values.\\
(i) Find an expression for $k$ in terms of $\theta$.\\
(ii) Show that $\mathrm { E } ( X ) = \frac { 1 } { 3 } \theta$.
You are given that $\operatorname { Var } ( X ) = \frac { 2 } { 9 } \theta ^ { 2 }$. A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ of $n$ observations of $X$ is obtained. The estimator $T _ { 1 }$ is defined as $T _ { 1 } = \frac { 3 } { n } \sum _ { i = 1 } ^ { n } X _ { i }$.\\
(iii) Show that $T _ { 1 }$ is an unbiased estimator of $\theta$, and find the variance of $T _ { 1 }$.\\
(iv) A second unbiased estimator $T _ { 2 }$ is defined by $T _ { 2 } = \frac { 1 } { 3 } \left( X _ { 1 } + 3 X _ { 2 } + 5 X _ { 3 } \right)$. For the case $n = 3$, which of $T _ { 1 }$ and $T _ { 2 }$ is more efficient?
\section*{OCR}
\hfill \mbox{\textit{OCR S4 2014 Q7 [12]}}