The joint probability distribution of the number of action movies (X) and the number of science fiction movies (Y ) that are released during the holiday period is given in the following table. y 0 1 2 x 3 1/12 1/9 5/36 4 1/8 1/6 1/9 5 1/24 1/18 1/6 (a) [1 mark] Find the probability that 3 action movies are released this holiday period. (b) [2 marks] Find the probability that 3 action movies are released this holiday period, given that 1 science fiction movie was released this holiday period. (c) [2 marks] Are the number of action movies released and the number of science fiction movies released independent variables? Explain why or why not. (d) [4 marks] Calculate the covariance between the number of action movies released and the number of science fiction movies released. For movie studios to maximise their profits, there is supposedly a particular ratio that helps determine how many action movies should be released, relative to science fiction movies, in any given period. (e) [4 marks] Calculate the expected value of the ratio XY X+Y