What is the p.d.f of $Z$?

Question

Let $f(x)=\frac{1}{2}\sin x \; (0<x<\pi )$ be the p.d.f of $X$ and $g(x)=\begin{cases} x & \text{ if } x<\frac{\pi }{3} \\ \frac{\pi }{9} & \text{ if } x\geqslant \frac{\pi }{3} \end{cases}$ and $Z=g(X)$. What is the p.d.f. of $Z$? Since $g$ is not injective on $x\geq \frac{\pi }{3}$, I struggle with this problem. please help me..

Answer 1

In the range $(0<X<\frac{\pi}{3})$ the density is the same $f_X$. When $X\geq \frac{\pi}{3}$ the rv is discrete, and in the point $Z=\frac{\pi}{9}$ the rv cumulates a positive probability mass...

Thus the requested "mixed" pdf is

$$ f_Z(z) = \begin{cases} \frac{sin (z)}{2}, & \text{if $0<z<\frac{\pi}{9} \cup \frac{\pi}{9}<z<\frac{\pi}{3} $ } \\ \frac{3}{4}, & \text{if $z=\frac{\pi}{9}$}\\ 0, & \text{elsewhere} \end{cases}$$