Question 1 (10 marks) a) A discrete random variable Y has the following probability function (or proba- bility mass function): i; if y = 1 m. 1; if y = 2 M9) = H" = y) = m i; if y = 3 m. 0; otherwise (i) Show that m = 6 if fy) is a valid probability function. (2 marks) (ii) Calculate the expected value of Y and the expected value of Y2 . (4 marks] (iii) Hence or otherwise, calculate the variance of Y. (1 mark) b) Let Z=3Y+2, where Y is dened inPart (a). (i) State the sample space of Z. {1 mark) (ii) Calculate the expected value of Z. {1 mark) (iii) Calculate the variance of Z. (1 mark)