Consider the random variables x, y with joint distribution y \ x 0 1 2 0 1/9 2/9

Consider the random variables x, y with joint distribution y \ x 0 1 2 0 1/9 2/9 1/9 1 2/9 0 2/9 2 0 1/9 0 1. (1 pts) find the marginal distributions of this joint distribution (i.e., compute the probability functions, p(x = x) and p(y = x), for all possible x). 2. (1 pts) find the conditional distribution of x given y = 1 (i.e., compute the probability function, p(x = x | y = 1), for all possible x). 3. (1 pts) are x and y independent? if yes, show that one of the equivalent characterizations of independence (given in class) is satisfied. if no, show that one of these characterizations is not satisfied.