| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2009 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | CDF to PDF derivation |
| Difficulty | Standard +0.3 This is a straightforward S3 question requiring standard techniques: (i) solving F(u) = 0.75 for the upper quartile involves basic logarithms, and (ii) differentiating the CDF piecewise to find the PDF is a routine procedure. Both parts are direct applications of definitions with no conceptual challenges or problem-solving required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
2 The continuous random variable $U$ has (cumulative) distribution function given by
$$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0 \\ 1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$
(i) Find the upper quartile of $U$.\\
(ii) Find the probability density function of $U$.
\hfill \mbox{\textit{OCR S3 2009 Q2 [5]}}