| Date | May 2010 | Marks available | 5 | Reference code | 10M.1.hl.TZ2.1 |
| Level | HL only | Paper | 1 | Time zone | TZ2 |
| Command term | Determine and Find | Question number | 1 | Adapted from | N/A |
Question
A continuous random variable X has the probability density function f given by
\[f(x) = \left\{ {\begin{array}{*{20}{c}}
{c(x - {x^2}),}&{0 \leqslant x \leqslant 1} \\
{0,}&{{\text{otherwise}}{\text{.}}}
\end{array}} \right.\]
(a) Determine c.
(b) Find \({\text{E}}(X)\).
Markscheme
(a) the total area under the graph of the pdf is unity (A1)
area \( = c\int_0^1 {x - {x^2}{\text{d}}x} \)
\( = c\left[ {\frac{1}{2}{x^2} - \frac{1}{3}{x^3}} \right]_0^1\) A1
\( = \frac{c}{6}\)
\( \Rightarrow c = 6\) A1
(b) \({\text{E}}(X) = 6\int_0^1 {{x^2} - {x^3}{\text{d}}x} \) (M1)
\( = 6\left( {\frac{1}{3} - \frac{1}{4}} \right) = \frac{1}{2}\) A1
Note: Allow an answer obtained by a symmetry argument.
[5 marks]
Examiners report
Most candidates made a meaningful attempt at this question with many gaining the correct answers. One or two candidates did not attempt this question at all.
