| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Topic | Discrete Random Variables |
| Type | Optimal estimator construction |
| Difficulty | Challenging +1.8 This is a multi-part statistics question requiring derivation of order statistic distributions and construction of unbiased estimators. Part (i) involves routine expectation calculation. Part (ii) requires understanding that P(M≤x) = P(all Xi≤x) = [F(x)]³, then differentiating to get the pdf—this is A-level accessible but requires careful reasoning about order statistics. Part (iii) applies the same expectation technique. While conceptually sophisticated for A-level (order statistics and estimator construction), the calculus is straightforward and the question provides significant scaffolding. This is harder than typical A-level but not exceptionally so for Further Maths/Pre-U. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.05b Unbiased estimates: of population mean and variance |
(i) $\int_0^k x\cdot\dfrac{3x^2}{k^3}\,\mathrm{d}x = \tfrac{3}{4}k$
$\text{E}(\tfrac{4}{3}X) = k$, so $\tfrac{4}{3}X$ unbiased **AG**
- M1: Attempt $\int xf(x)$, correct limits
- A1: $\tfrac{3}{4}k$, ae exact f
- A1: Must state "unbiased"
**[3]**
(ii) $P(X \leq x) = \int_0^x \dfrac{3x^2}{k^3}\,\mathrm{d}x = \left(\dfrac{x^3}{k^3}\right)$
$P(M \leq m) = \left(\dfrac{x^3}{k^3}\right)^3 = \dfrac{x^9}{k^9}$
$f_M(x) = \dfrac{9x^8}{k^9}$ **AG**
- B1: Needs convincing derivation
- M1: $[F_X(x)]^3$
- M1: Differentiate
- A1: Full derivation of AG. Ignore other ranges
**[4]**
(iii) $\int_0^k x\cdot\dfrac{9x^8}{k^9}\,\mathrm{d}x = \tfrac{9}{10}k$
Hence $E_2 = \tfrac{10}{9}M$
- M1: Attempt $\int xf_M(x)$, ignore limits
- A1: Correct $\text{E}(M)$
- A1ft: If $\text{E}(M) = kc$, allow $M/c$
**[3]**7 A continuous random variable $X$ has probability density function
$$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { k ^ { 3 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a parameter.\\
(i) Find $\mathrm { E } ( X )$. Hence show that $\frac { 4 } { 3 } X$ is an unbiased estimator of $k$.
Three independent observations of $X$ are denoted by $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$, and the largest value of $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$ is denoted by $M$.\\
(ii) Write down an expression for $\mathrm { P } ( M \leqslant x )$, and hence show that the probability density function of $M$ is
$$f _ { M } ( x ) = \begin{cases} \frac { 9 x ^ { 8 } } { k ^ { 9 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } . \end{cases}$$
(iii) Find $\mathrm { E } ( M )$ and use your answer to construct an unbiased estimator of $k$ based on $M$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q7 [10]}}