Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
The random variable $X$ has probability density function $f$. Its graph, $y = f(x)$, is a semicircle centred at $(0,0)$ with radius $\sqrt{\tfrac{2}{\pi}}$, and it lies entirely above the $x$-axis. For all other values, $f(x) = 0$ (see diagram).
(a)[2]
Verify that $f$ is a probability density function.
(b)[6]
A and B are the points at which the line $x = \sqrt{\tfrac{1}{\pi}}$ intersects the $x$-axis and the semicircle, respectively. Show that angle $AOB$ is $\tfrac{1}{4}\pi$ radians and then find $\mathrm{P}\left(X > \sqrt{\tfrac{1}{\pi}}\right)$.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Correct evaluation $\frac{1}{2}\pi\left(\sqrt{\frac{2}{\pi}}\right)^2$” …