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.