| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Showing estimator is unbiased |
| Difficulty | Standard +0.3 This is a straightforward statistics question requiring standard formulas for uniform distribution (mean = (a+b)/2, variance = (b-a)²/12) and basic properties of estimators. Part (a) is direct substitution, part (b) requires recognizing that E[X̄] = μ and using the relationship between sample mean variance and population variance. While it's Further Maths content, the techniques are routine applications of known results with no novel problem-solving required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \theta + 2\) | B1 | — |
| \(\text{Var}(X) = 3\) | B1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(\bar{X}) = \theta + 2\) OR \(E(\bar{X} - 2) = \theta\) | M1 | FT their linear \(E(X)\) for M1A1 |
| \(\bar{X} - 2 \text{ is an unbiased estimator for } \theta\) | A1 | — |
| \(\text{SE}(\bar{X} - 2) = \text{SE}(\bar{X})\) | M1 | Used FT their \(\text{Var}(X)\) |
| \(= \sqrt{\frac{3}{9}} = \sqrt{\frac{\sqrt{3}}{3}}\) oe | A1 | — |
## (a)
$E(X) = \theta + 2$ | B1 | —
$\text{Var}(X) = 3$ | B1 | —
## (b)
$E(\bar{X}) = \theta + 2$ OR $E(\bar{X} - 2) = \theta$ | M1 | FT their linear $E(X)$ for M1A1
$\bar{X} - 2 \text{ is an unbiased estimator for } \theta$ | A1 | —
$\text{SE}(\bar{X} - 2) = \text{SE}(\bar{X})$ | M1 | Used FT their $\text{Var}(X)$
$= \sqrt{\frac{3}{9}} = \sqrt{\frac{\sqrt{3}}{3}}$ oe | A1 | —
**Total: [6]**
---The continuous random variable $X$ is uniformly distributed over the interval $(\theta - 1, \theta + 5)$, where $\theta$ is an unknown constant.
\begin{enumerate}[label=(\alph*)]
\item Find the mean and the variance of $X$. [2]
\item Let $\overline{X}$ denote the mean of a random sample of 9 observations of $X$. Find, in terms of $\overline{X}$, an unbiased estimator for $\theta$ and determine its standard error. [4]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2019 Q2 [6]}}