I randomly pick two integers from 1 to n with replacement where n is a positive integer. (a) What is the probability that the first number will be greater than the second? Derive a simplified expressions in terms of n. (b) Let the random variable X be the maximum of the two numbers. Find the probability mass function of X and express as a formula. (c) Find E(X) and simplify as much as possible (use formulas for the sum and sum of squares of the first n integers which you can find online). (d) Do a simulation in R to verify your answers in (a), (b) and (c) for n = 4 and n = 15. Note: you can do this simulation without a loop. See the pmax function.

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