(Solution) - For the situation of Example 1 2 20 the average of the -(2025 Original AI-Free Solution)

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For the situation of Example 1.2.20, the average of the original set of numbers {2,4,9,12} is 22, which has the highest probability.
(a) Prove that, in general, if we sample with replacement from the set {x1 , x2,...., xn}, the outcome with average (x1+x2+ . . .+ xn )/n is the most likely, having probability

(b) Use Stirling's Formula (Exercise 1.28) to show that n!/nn ??n2?/en (Hall 1992, Appendix I).
(c) Show that the probability that a particular x* is missing from an outcome is