(Solved): A continuous random variable \( \mathrm{Z} \) has the probability density function given by: \[ f( ...

A continuous random variable \( \mathrm{Z} \) has the probability density function given by: \[ f(z)=\left\{\begin{array}{ll} k \sin \frac{1}{2} z, & 0 \leq z \leq \frac{2 \pi}{3} \\ k \sin \left(z-\frac{\pi}{3}\right), & \frac{2 \pi}{3} \leq z \leq \frac{4 \pi}{3} \\ 0, & \text { elsewhere } \end{array}\right. \] (i) Calculate the value of \( k \) and sketch the graph of \( f(z) \). [4 marks] (ii) Obtain the cumulative distribution function of \( Z \) and sketch its graph. [5 marks] (iii) Calculate the median of \( Z \). [2 marks]