(a)[3]
Show that $P\left(X < \tfrac{1}{2}\right) = \tfrac{1}{2} + \tfrac{1}{\pi}$, as claimed.
(b)[5]
Show that $\mathrm{E}(X) = \tfrac{1}{2} - \tfrac{2}{\pi^2}$, as required.
Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
$X$ is a random variable with probability density function defined as
$f(x) = \begin{cases} 1 + \cos \pi x, & 0 \leq x \leq 1, \\ 0, & \text{otherwise}. \end{cases}$