| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2009 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moment generating functions |
| Type | Show unbiased estimator |
| Difficulty | Challenging +1.8 This is a Further Maths Statistics question requiring understanding of MGFs, unbiased estimators, and efficiency. Part (i) is routine integration, but parts (ii)-(iv) require careful algebraic manipulation of expectations and variance calculations for linear combinations of random variables. The efficiency comparison demands computing and comparing variances of two complex estimators, which is non-trivial but follows standard S4 techniques without requiring novel insight. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain derivative \(k(37 + 10y - 2y^2)^{-\frac{1}{2}}f(y)\) | M1 | any constant \(k\); any linear function for \(f\) |
| Obtain \(\frac{1}{4}(10 - 4y)(37 + 10y - 2y^2)^{-\frac{1}{2}}\) | A1 2 | or equiv |
| Answer | Marks | Guidance |
|---|---|---|
| Either: Sub'te \(y = 3\) in expression for \(\frac{dx}{dy}\) | *M1 | |
| Take reciprocal of expression/value | *M1 | and without change of sign |
| Obtain \(-7\) for gradient of tangent | A1 | |
| Attempt equation of tangent | M1 | dep *M *M |
| Obtain \(y = -7x + 52\) | A1 5 | and no second equation |
| Or: Sub'te \(y = 3\) in expression for \(\frac{dx}{dy}\) | M1 | |
| Attempt formation of eq'n \(x = m'y + c\) | M1 | where \(m'\) is attempt at \(\frac{dy}{dx}\) |
| Obtain \(x = -7 - \frac{1}{2}(y - 3)\) | A1 | or equiv |
### (i)
Obtain derivative $k(37 + 10y - 2y^2)^{-\frac{1}{2}}f(y)$ | M1 | any constant $k$; any linear function for $f$ |
Obtain $\frac{1}{4}(10 - 4y)(37 + 10y - 2y^2)^{-\frac{1}{2}}$ | A1 2 | or equiv |
### (ii)
Either: Sub'te $y = 3$ in expression for $\frac{dx}{dy}$ | *M1 | |
Take reciprocal of expression/value | *M1 | and without change of sign |
Obtain $-7$ for gradient of tangent | A1 |
Attempt equation of tangent | M1 | dep *M *M |
Obtain $y = -7x + 52$ | A1 5 | and no second equation |
Or: Sub'te $y = 3$ in expression for $\frac{dx}{dy}$ | M1 |
Attempt formation of eq'n $x = m'y + c$ | M1 | where $m'$ is attempt at $\frac{dy}{dx}$ |
Obtain $x = -7 - \frac{1}{2}(y - 3)$ | A1 | or equiv |
Attempt rearrangement to6 The continuous random variable $X$ has probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} 0 & x < a , \\ \mathrm { e } ^ { - ( x - a ) } & x \geqslant a , \end{cases}$$
where $a$ is a constant. $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are $n$ independent observations of $X$, where $n \geqslant 4$.\\
(i) Show that $\mathrm { E } ( X ) = a + 1$.\\
$T _ { 1 }$ and $T _ { 2 }$ are proposed estimators of $a$, where
$$T _ { 1 } = X _ { 1 } + 2 X _ { 2 } - X _ { 3 } - X _ { 4 } - 1 \quad \text { and } \quad T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { 4 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 2 ( n - 2 ) } - 1 .$$
(ii) Show that $T _ { 1 }$ and $T _ { 2 }$ are unbiased estimators of $a$.\\
(iii) Determine which is the more efficient estimator.\\
(iv) Suggest another unbiased estimator of $a$ using all of the $n$ observations.
\hfill \mbox{\textit{OCR S4 2009 Q6 [13]}}