(Solved): can you solve q2 please? 2. Let \( X \) and \( Y \) be discrete random variables, which are independ ...

2. Let \( X \) and \( Y \) be discrete random variables, which are independent of each other, with probability mass functions given by \[ \mathbf{P}(X=k)=\left\{\begin{array}{ll} \left(\frac{1}{2}\right)^{k}, & k=1,2,3, \ldots \\ 0, & \text { otherwise, } \end{array} \quad \mathbf{P}(Y=k)=\left\{\begin{array}{ll} c\left(\frac{2}{3}\right)^{k}, & k=2,3, \ldots \\ 0, & \text { otherwise } \end{array}\right.\right. \] Let \( Z=\min (X, Y) \). (i) Prove that \( c=\frac{3}{4} \). [5 marks] (ii) For \( k \in\{1,2, \ldots\} \) find \( \mathbf{P}(X>k) \) and \( \mathbf{P}(Y>k) \). [5 marks] (iii) For \( k \in\{1,2, \ldots\} \) find \( \mathbf{P}(Z>k) \). [5 marks] (iv) Hence, or otherwise, find the probability mass function of \( Z \). [5 marks]