2. Sample Maximum a) Let N be a fixed positive integer. Let X be a random variable that has possible values {1, 2, , N}. Consider the probabilities F (m) = P(X g m) for m 2 1. It's a good idea to draw a number line and color the event {X S m} for a generic m. For 1 S k S N , write P(X = k) in terms of the values F (m) for m 2 1. If you get stuck. take a look at Example 2.2.2 in the textbook. b) Let X1, X2, , X\" be the results of )1 draws made at random with replacement from {1, 2, , N}. Let M = max{X1, X2, , Xn}. Use the method developed in Part a to find the distribution of M. [Think about how M can be at most m. For this to happen, how big can X1 be? What about X2? If you have trouble starting out in the general case, pick some small numbers like N = 10, m = 4, and n = 3 to see what's going on.] c) Now let X1, X2, . .. , X\" be the results of It draws made at random without replacement from {1, 2, , N}. You can assume n S N in this case. Let M = max{X1, X2, , n}. Use the method developed in Part a to find the distribution of M. Start by carefully specifying the possible values of M