| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find mode of distribution |
| Difficulty | Moderate -0.3 This is a straightforward probability density function question requiring integration to find k (using standard sine integral) and identifying the mode from a sketch. The integration is routine, and the mode is simply the maximum of the sine function. Slightly easier than average due to standard techniques and no complex problem-solving required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
1 The lifetime, $T$ years, of a mortgage may be modelled by the random variable $T$ with probability density function $\mathrm { f } ( t )$, where
$$\mathrm { f } ( t ) = \begin{cases} k \sin \left( \frac { 3 } { 32 } t \right) & 0 \leqslant t \leqslant 8 \pi \\ 0 & \text { otherwise } \end{cases}$$
(i) Show that $k = \frac { 3 } { 32 } ( 2 - \sqrt { 2 } )$.\\
(ii) Sketch the graph of $\mathrm { f } ( t )$ and state the mode.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q1}}