You observe k i.i.d. copies of the discrete uniform random variable Xi , which takes values 1 through n with equal probability.

Question

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∈{1,2,…,n} .

2.) Find the probability that M=1 .

3.) Find the probability that M=m for m∈{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 k→∞ , E[M]→

Answer 1

1. (m/n)^k