SPM.1.AHL.TZ0.7

pestleMathematicsAAHLPaper 1SPM· ahl-4-14-properties-of-discrete-and-continuous-random-variablessource ↗

A continuous random variable X has the probability density function $f$ given by

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\pi x}}{{36}}{\text{sin}}\left( {\frac{{\pi x}}{6}} \right){\text{,}}}&{0 \leqslant x \leqslant 6} \\ {0{\text{,}}}&{{\text{otherwise}}} \end{array}} \right.$ .

Find P(0 ≤ X ≤ 3).

Markscheme / solution

attempting integration by parts, eg

$u = \frac{{\pi x}}{{36}}{\text{,}}\,{\text{d}}u = \frac{\pi }{{36}}{\text{d}}x{\text{,}}\,{\text{d}}v = {\text{sin}}\left( {\frac{{\pi x}}{6}} \right){\text{d}}x{\text{,}}\,v = - \frac{6}{\pi }{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)$ (M1)

P(0 ≤ X ≤ 3) $= \frac{\pi }{{36}}\left( {\left[ { - \frac{{6x}}{\pi }{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)} \right]_0^3 + \frac{6}{\pi }\int\limits_0^3 {{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)} {\text{d}}x} \right)$ (or equivalent) A1A1

Note: Award A1 for a correct $u v$ and A1 for a correct $\int {v\,{\text{d}}u}$ .

attempting to substitute limits M1

$\frac{\pi }{{36}}\left[ { - \frac{{6x}}{\pi }{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)} \right]_0^3 = 0$ (A1)

so P(0 ≤ X ≤ 3) $= \frac{1}{\pi }\left[ {{\text{sin}}\left( {\frac{{\pi x}}{6}} \right)} \right]_0^3$ (or equivalent) A1

$= \frac{1}{\pi }$ A1

[7 marks]

Examiners’ report

[N/A]