Suppose X1 is the value on the upper face of one six-sided die, and X2 is the value on the upper face of a second
six-sided die. The joint distribution of X1 and X2 is a full specification of the probabilities P(X1 = x1, X2 = x2) for all possible values x1 of X1 and x2 of X2. (Note that the event (X1 = x1, X2 = x2) is the intersection of the events X1 = x1 and X2 = x2.) Suppose X1 and X2 are independent random variables; recall we defined in lecture that two such random variables are independent if and only if P(X1 = x1, X2 = x2) = P(X1 = x1)P(X2 = x2) Finally, let M be the minimum-that is, the smaller, of the numbers on the upper faces of the two die.
Compute 1 (a) The joint distribution of X1 and X2;
(b) The joint distribution of M and X2. Determine whether or not M and X2 are independent and justify your answer.