| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find or specify CDF |
| Difficulty | Standard +0.3 This is a straightforward S3 question involving standard techniques: integrating a simple power function to find the CDF, using the transformation formula for PDFs (with clear guidance), and computing an expectation. All steps are routine applications of A-level methods with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\int_1^{\infty} \frac{3}{x^4} dx\) | M1 | Any variable |
| Answer | Marks | Guidance |
|---|---|---|
| A1 | 2 | |
| (ii) \(G(y) = P(Y \le y) = P(T \le y^{1/3}) = F(y^{1/3}) = 1 - 1/y\) | M1, A1, M1, A1√ | ft \(F(t)\) |
| \(g(y) = G'(y) = 1/y^2\), \(y \ge 1\) AG | M1, A1 | 6 |
| (iii) EITHER: \(\int_1^{\infty} \frac{\sqrt{y}}{y^2} dy\) OR: \(\int_1^{\infty} \frac{3t^2}{t^4} dt\) | M1 | |
| \(\left[-2y^{-1/2}\right]_1^{\infty}\) or \(\left[-2t^{-3/2}\right]_1^{\infty}\); \(= 2\) | B1, A1 | 3 |
**(i)** $\int_1^{\infty} \frac{3}{x^4} dx$ | M1 | Any variable
$$F(t) = \begin{cases}
1 - \frac{1}{t^3} & t \ge 1 \\
0 & \text{otherwise}
\end{cases}$$
| A1 | 2
**(ii)** $G(y) = P(Y \le y) = P(T \le y^{1/3}) = F(y^{1/3}) = 1 - 1/y$ | M1, A1, M1, A1√ | ft $F(t)$
$g(y) = G'(y) = 1/y^2$, $y \ge 1$ AG | M1, A1 | 6
**(iii)** EITHER: $\int_1^{\infty} \frac{\sqrt{y}}{y^2} dy$ OR: $\int_1^{\infty} \frac{3t^2}{t^4} dt$ | M1 |
$\left[-2y^{-1/2}\right]_1^{\infty}$ or $\left[-2t^{-3/2}\right]_1^{\infty}$; $= 2$ | B1, A1 | 36 The lifetime of a particular machine, in months, can be modelled by the random variable $T$ with probability density function given by
$$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { t ^ { 4 } } & t \geqslant 1 \\ 0 & \text { otherwise. } \end{cases}$$
(i) Obtain the (cumulative) distribution function of $T$.\\
(ii) Show that the probability density function of the random variable $Y$, where $Y = T ^ { 3 }$, is given by $\mathrm { g } ( y ) = \frac { 1 } { y ^ { 2 } }$, for $y \geqslant 1$.\\
(iii) Find $\mathrm { E } ( \sqrt { Y } )$.
\hfill \mbox{\textit{OCR S3 2007 Q6 [11]}}