You observe k i.i.d. copies of the discrete uniform random variable Xi, which takes values 1 through n with equal probability.
Define the random variable M as the maximum of these random variables, M=maxi(Xi).
1) Find the probability that M ⤠m, as a function of m, for m â {1,2,...,n}
2) Find the probability that M=1.
3) Find the probability that M=m for m â {2,3,...n}.
4) For n=2, find E[M] and Var(M) as a function of k.
5) As k (the number of samples) becomes very large, what is E[M] in terms of n?
As kâ[infinity], E[M]â