(Solved): Suppose we roll a tetrahedral die with sides labeled one through four. Then, for a single roll, = ...

Suppose we roll a tetrahedral die with sides labeled one through four. Then, for a single roll, ? = {1, 2, 3, 4}. Assume the die is fair, so each of the events {1}, . . . , {4} has probability 1/4. (a) As in Example 5, let A be the event that we get an even number and B be the event that we get a number divisible by three. Are A and B independent? 9 (b) As in Example 8, let X = 1A and Y = 1B. What distribution does X have? Calculate E[X] and Var(X). What distribution does Y have? Calculate E[Y ] and Var(Y ). Write down the joint distribution of X and Y by filling in the probability of
each cell in the table below.

(b) As in Example 8 , let \( X=\mathbf{1}_{A} \) and \( Y=\mathbf{1}_{B} \). What distribution does \( X \) have? Calculate \( \mathbb{E}[X] \) and \( \operatorname{Var}(X) \). What distribution does \( Y \) have? Calculate \( \mathbb{E}[Y] \) and \( \operatorname{Var}(Y) \) (c) Write down the joint distribution of \( X \) and \( Y \) by filling in the probability of each cell in the table below. (d) Add the columns of the table to get the marginal distribution of \( X \) and add the rows to get the marginal distribution of \( Y \). (e) What would the table look like if \( X \) and \( Y \) had the same marginal distributions but were independent?