(Solved): can you solve q3 please? 3. Let \( X \) and \( Y \) be continuous random variables, which are indepe ...

3. Let \( X \) and \( Y \) be continuous random variables, which are independent of each other, with probability density functions given by \[ f_{X}(x)=\left\{\begin{array}{ll} \mathrm{e}^{-x}, & x>0 \\ 0, & \text { otherwise, } \end{array} \quad f_{Y}(y)=\left\{\begin{array}{ll} y \mathrm{e}^{-y}, & y>0 \\ 0, & \text { otherwise. } \end{array}\right.\right. \] (i) Find the probability density function of \( X+Y \). [6 marks] (ii) Let \( Z=X / Y \). Find the joint probability density function of \( (Z, Y) \). [8 marks] (iii) Hence, or otherwise, find the probability density of \( Z \). [6 marks]