| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | CDF of transformed variable |
| Difficulty | Standard +0.8 This S3 question requires understanding of CDF transformations and the relationship between CDFs and PDFs. Part (i) involves careful manipulation of inequalities when transforming variables (non-trivial since Y is a decreasing function of V), requiring students to show a specific form and determine the valid interval. Part (ii) requires finding E(1/Y²) which involves recognizing this equals E((1+V)²) and either integrating or using the PDF of Y cleverly. While systematic, this goes beyond routine S3 exercises and requires solid understanding of transformation techniques. |
| Spec | 5.03g Cdf of transformed variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(8y^3\) obtained correctly | M1, A1 | Correct range. Condone omission of \(y \leq 0\) |
| \(g(y) = \begin{cases} 24y^2 & 0 < y \leq 1/2 \\ 0 & \text{otherwise} \end{cases}\) | M1, A1 | For \(G(y)\) - Correct answer with range |
| \(\int 24y^2/y^3 \, dy\) with limits \(= 12\) | M1, A1 | With attempt at integration |
## 4(i)
$G(y) = P(Y \leq y) = P(1/(1+V) \leq y) = P(V \geq 1/y - 1) = 1 - F(1/y - 1)$
$$G(y) = \begin{cases} 0 & y \leq 0 \\ 8y^3 & 0 < y \leq 1/2 \\ 1 & y > 1/2 \end{cases}$$
| $8y^3$ obtained correctly | M1, A1 | Correct range. Condone omission of $y \leq 0$ |
| --- | --- | --- |
| $g(y) = \begin{cases} 24y^2 & 0 < y \leq 1/2 \\ 0 & \text{otherwise} \end{cases}$ | M1, A1 | For $G(y)$ - Correct answer with range |
| $\int 24y^2/y^3 \, dy$ with limits $= 12$ | M1, A1 | With attempt at integration |
**Total: [9]**
---4 The continuous random variable $V$ has (cumulative) distribution function given by
$$\mathrm { F } ( v ) = \begin{cases} 0 & v < 1 \\ 1 - \frac { 8 } { ( 1 + v ) ^ { 3 } } & v \geqslant 1 \end{cases}$$
The random variable $Y$ is given by $Y = \frac { 1 } { 1 + V }$.\\
(i) Show that the (cumulative) distribution function of $Y$ is $8 y ^ { 3 }$, over an interval to be stated, and find the probability density function of $Y$.\\
(ii) Find $\mathrm { E } \left( \frac { 1 } { Y ^ { 2 } } \right)$.
\hfill \mbox{\textit{OCR S3 2010 Q4 [9]}}