(a)[2]
Hence, show that $a = \dfrac{2}{\sqrt{\pi}}$.
(b)[2]
Hence, show that $f(x)=\sqrt{\dfrac{4}{\pi}-x^2}$.
(c)[4]
Hence, show that $\mathrm{E}(X) = \frac{8}{3\sqrt{\pi}}$.
Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
The diagram depicts the probability density function, $f$, for a random variable $X$. It is a quarter circle that lies wholly in the first quadrant, has centre $(0,0)$ and radius $a$, where $a$ is a positive constant. For all other $x$, $f(x)=0$.