The probability density function of the random variable X is defined as

\[f(x) = \left\{ {\begin{array}{*{20}{c}}
  {\sin x,}&{0 \leqslant x \leqslant \frac{\pi }{2}} \\
  {0,}&{{\text{otherwise}}{\text{.}}}
\end{array}} \right.\]

Find \({\text{E}}(X)\).

\(\int_0^{\frac{\pi }{2}} {x\sin x{\text{d}}x} \)     M1

\( = [ - x\cos x]_0^{\frac{\pi }{2}} + \int_0^{\frac{\pi }{2}} {\cos x{\text{d}}x} \)     M1(A1)

Note: Condone the absence of limits or wrong limits to this point.

\( = [ - x\cos x + \sin x]_0^{\frac{\pi }{2}}\)     A1

\( = 1\)     A1

[5 marks]

It was pleasing to note how many candidates recognised the expression that needed to be integrated and successfully used integration by parts to reach the correct answer.