4. (From the 2009 MATH2801/2901 final exam) A soccer club has two expensive star players recruited from overseas. Detailed match records give the bivariate distribution for (X, Y ) below, where X is the number of points obtained in the game (2 for a win, 1 for a draw, 0 for a loss), and Y is the number of stars playing in the game. The table below gives values of P(X = x, Y = y).

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4. (From the 2009 MATH2801 / 2901 ﬁnal exam) A soccer club has two expensive star players recruited from overseas. Detailedmatch records give the bivariate distribution for (X , Y) below, where X is thenumber of points obtained in the game (2 for a win, 1 for a draw, 0 for a loss), and Y is the number of stars playing in the game. The table below gives values of P(X = :1}, Y = y). y0 1 2 0 0.15 0.12 0.06a: 1 0.09 0.12 0.092 0.06 0.16 0.15 i) Jono goes to a game hoping to see the club win the game and hoping tosee at least one of the stars playing in the game. What is the chance that what Jono is hoping to see will actually happen? ii) a) Find the marginal distribution of X, the number of points obtainedin a game.b) Hence find the expected number of points obtained in a game.c) Find the variance of the number of points obtained in a game.iii) To Jono’s delight, when he arrives at the game he ﬁnds out that both stars will be playing in the game. What is the conditional distribution ofpoints obtained in the game, given that both stars are playing?