Question

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).

Problem 1(a)
Find the probability that M≤m, as a function of m, for m∈{1,2,…,n}.

Problem 1(b)
Find the probability that M=1.

Problem 1(c)
Find the probability that M=m for m∈{2,3,…n}.

Problem 1(d)
For n=2, find E[M] and Var(M) as a function of k.

E[M]=
Var[M]=

Problem 1(e)
As k (the number of samples) becomes very large, what is E[M] in terms of n?

As k→∞, E[M]→