1. Two games, X and Y, can be simultaneously played at a casino, where each game results in either $1.00 won, lost, or neither. Their probabilities, including xed row and column marginal probabilities, are shown in the tabulated joint probability mass function below, for some parameters or, [3, and y . (a) What condition(s) on a, , and 5V must hold if this is indeed a valid joint pmf. Explain (3 pts) (b) Compute the expected amount of the rst game X. Show all work. (3 pts) (c) Determine the variance of the rst game X. Show all work. (4 pts) ((1) Compute the expected amount of the second game Y. Show all work. (3 pts) (e) Determine the variance of the second game Y. Show all work. (4 pts) (f) Compute the covariance betweenX and Y, and Show that it only depends on y Show all work. (4 pts) (g) What condition(s) on a, , and 3/ must hold if the covariance is equal to zero? Eagplaiu. (2 pts) (h) What condition(s) on 0:, , and y must hold if X and Y statistically independent? EJquain. (2 pts)