| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2016 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Distribution of order statistics |
| Difficulty | Challenging +1.8 This S4 Further Maths question requires understanding of order statistics, deriving PDFs from CDFs, and constructing unbiased estimators. While conceptually demanding (involving P(S>s) = P(Y₁>s)P(Y₂>s), differentiation, and expectation calculations), it's a standard multi-part question with clear signposting ('show that', 'hence find'). Part (iv) tests conceptual understanding of efficiency. The algebraic manipulation is moderate, making this harder than typical A-level but within expected S4 range. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(S \leq s) = P(\text{at least one of } Y_1, Y_2 < s) = P(\text{not both } Y_1, Y_2 > s)\) | M1 | |
| \(1 - \left[1 - \left(\dfrac{a^5}{s^5}\right)\right]^2\) | \(\left(\dfrac{a^5}{s^5}\right)^2\) | |
| \(P(S > s) = \dfrac{a^{10}}{s^{10}}\) AG | A1 | cwo |
| CDF of \(S = 1 - a^{10}s^{-10}\); and differentiate | B1;M1 | |
| \(10a^{10}s^{-11}\) | A1 | |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(S) = \int_a^{\infty} s \cdot 10a^{10}s^{-11}\, ds\) | M1 | |
| \(\dfrac{10}{9}a\) | A1 | |
| \(\neq a\) | M1 | Must have ans \(ka\) for \(E(S)\), provided \(k>0\) |
| \(\dfrac{9}{10}S\) | B1ft | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(y) = 5a^5y^{-6}\) | B1 | |
| \(E(Y) = \int_a^{\infty} y \cdot 5a^5 y^{-6}\, dy\) | M1 | |
| \(= \dfrac{5}{4}a\) | A1 | |
| \(k = \dfrac{2}{5}\) | B1ft | \(1 \div (2 \times \text{coeff of } a)\). Must follow from an attempt at integration |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Find which of \(T_1\) and \(T_2\) has smaller variance | B1 | |
| [1] |
# Question 7:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(S \leq s) = P(\text{at least one of } Y_1, Y_2 < s) = P(\text{not both } Y_1, Y_2 > s)$ | M1 | |
| $1 - \left[1 - \left(\dfrac{a^5}{s^5}\right)\right]^2$ | | $\left(\dfrac{a^5}{s^5}\right)^2$ |
| $P(S > s) = \dfrac{a^{10}}{s^{10}}$ AG | A1 | cwo |
| CDF of $S = 1 - a^{10}s^{-10}$; and differentiate | B1;M1 | |
| $10a^{10}s^{-11}$ | A1 | |
| **[5]** | | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(S) = \int_a^{\infty} s \cdot 10a^{10}s^{-11}\, ds$ | M1 | |
| $\dfrac{10}{9}a$ | A1 | |
| $\neq a$ | M1 | Must have ans $ka$ for $E(S)$, provided $k>0$ |
| $\dfrac{9}{10}S$ | B1ft | |
| **[4]** | | |
## Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(y) = 5a^5y^{-6}$ | B1 | |
| $E(Y) = \int_a^{\infty} y \cdot 5a^5 y^{-6}\, dy$ | M1 | |
| $= \dfrac{5}{4}a$ | A1 | |
| $k = \dfrac{2}{5}$ | B1ft | $1 \div (2 \times \text{coeff of } a)$. Must follow from an attempt at integration |
| **[4]** | | |
## Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Find which of $T_1$ and $T_2$ has smaller variance | B1 | |
| **[1]** | | |
The image you've shared appears to be only the **back cover/contact page** of an OCR mark scheme document. It contains only OCR's contact information (address, phone numbers, email, website) and does not include any actual mark scheme content with questions, answers, or mark allocations.
To extract the mark scheme content, please share the **interior pages** of the document that contain the actual questions and marking criteria.7 A continuous random variable $Y$ has cumulative distribution function
$$\mathrm { F } ( y ) = \left\{ \begin{array} { c c }
0 & y < a \\
1 - \frac { a ^ { 5 } } { y ^ { 5 } } & y \geqslant a
\end{array} \right.$$
where $a$ is a parameter.\\
Two independent observations of $Y$ are denoted by $Y _ { 1 }$ and $Y _ { 2 }$. The smaller of them is denoted by S .\\
(i) Show that $P ( S > \mathrm { s } ) = \frac { a ^ { 10 } } { s ^ { 10 } }$ and hence find the probability density function of $S$.\\
(ii) Show that $S$ is not an unbiased estimator of $a$, and construct an unbiased estimator of $a , T _ { 1 }$ based on $S$.\\
(iii) Construct another unbiased estimator of $a , T _ { 2 }$, of the form $k \left( Y _ { 1 } + Y _ { 2 } \right)$, where $k$ is a constant to be found.\\
(iv) Without further calculation, explain how you would decide which of $T _ { 1 }$ and $T _ { 2 }$ is the more efficient estimator.
\hfill \mbox{\textit{OCR S4 2016 Q7 [14]}}