| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2013 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Optimal estimator construction |
| Difficulty | Challenging +1.2 This is a systematic S4 question on unbiased estimators requiring standard techniques: finding expectations, solving for constants using unbiasedness conditions, comparing efficiency via variance calculations, and extending to second moment estimators. While it involves multiple parts and careful algebraic manipulation, each step follows established procedures without requiring novel insight or particularly challenging problem-solving. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(T_1) = 2E(X) + 3E(Y) = 8\mu\) | M1, A1 | |
| Unbiased estimate \(= (X_1+X_2+Y_1+Y_2+Y_3)/8\) | A1 | NOT \(\frac{2x+3y}{8}\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(T_2) = 2\mu + 6c\mu = \mu \Rightarrow c = -\frac{1}{6}\) | M1, A1 | Setting up an equation |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var}(T_1) = (2\sigma^2+9\sigma^2)/64 = \frac{11}{64}\sigma^2\) | M1, A1 | Using var of sum = sum of var |
| \(\text{Var}(T_2) = \sigma^2+\sigma^2+\frac{1}{36}(3\sigma^2+3\sigma^2+3\sigma^2) = \frac{9}{4}\sigma^2\) | A1 | |
| \(T_1\) has the smaller variance so is more efficient | A1ft | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(T_3) = a(2\sigma^2+2\mu^2)+b(9\sigma^2+12\mu^2) = \sigma^2\) | M1A1 | \((\text{Var}(X)=)\ E(X^2)-[E(X)]^2\) seen or implied: M1 |
| Coefficient of \(\mu^2 = 0\) gives \(2a+12b = 0\) | ||
| Coefficient of \(\sigma^2 = 1\) gives \(2a+9b = 1\) | B1 | either equation |
| Solve to give \(a=2\) and \(b = -\frac{1}{3}\) | A1 | |
| [4] |
# Question 6:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(T_1) = 2E(X) + 3E(Y) = 8\mu$ | M1, A1 | |
| Unbiased estimate $= (X_1+X_2+Y_1+Y_2+Y_3)/8$ | A1 | NOT $\frac{2x+3y}{8}$ |
| **[3]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(T_2) = 2\mu + 6c\mu = \mu \Rightarrow c = -\frac{1}{6}$ | M1, A1 | Setting up an equation |
| **[2]** | | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var}(T_1) = (2\sigma^2+9\sigma^2)/64 = \frac{11}{64}\sigma^2$ | M1, A1 | Using var of sum = sum of var |
| $\text{Var}(T_2) = \sigma^2+\sigma^2+\frac{1}{36}(3\sigma^2+3\sigma^2+3\sigma^2) = \frac{9}{4}\sigma^2$ | A1 | |
| $T_1$ has the smaller variance so is more efficient | A1ft | |
| **[4]** | | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(T_3) = a(2\sigma^2+2\mu^2)+b(9\sigma^2+12\mu^2) = \sigma^2$ | M1A1 | $(\text{Var}(X)=)\ E(X^2)-[E(X)]^2$ seen or implied: M1 |
| Coefficient of $\mu^2 = 0$ gives $2a+12b = 0$ | | |
| Coefficient of $\sigma^2 = 1$ gives $2a+9b = 1$ | B1 | either equation |
| Solve to give $a=2$ and $b = -\frac{1}{3}$ | A1 | |
| **[4]** | | |
---6 The continuous random variable $X$ has mean $\mu$ and variance $\sigma ^ { 2 }$, and the independent continuous random variable $Y$ has mean $2 \mu$ and variance $3 \sigma ^ { 2 }$. Two observations of $X$ and three observations of $Y$ are taken and are denoted by $X _ { 1 } , X _ { 2 } , Y _ { 1 } , Y _ { 2 }$ and $Y _ { 3 }$ respectively.\\
(i) Find the expectation of the sum of these 5 observations and hence construct an unbiased estimator, $T _ { 1 }$, of $\mu$.\\
(ii) The estimator $T _ { 2 }$, where $T _ { 2 } = X _ { 1 } + X _ { 2 } + c \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)$, is an unbiased estimator of $\mu$. Find the value of the constant $c$.\\
(iii) Determine which of $T _ { 1 }$ and $T _ { 2 }$ is more efficient.\\
(iv) Find the values of the constants $a$ and $b$ for which
$$a \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } \right) + b \left( Y _ { 1 } ^ { 2 } + Y _ { 2 } ^ { 2 } + Y _ { 3 } ^ { 2 } \right)$$
is an unbiased estimator of $\sigma ^ { 2 }$.
\hfill \mbox{\textit{OCR S4 2013 Q6 [13]}}