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)

1a. Find the probability that 𝑀≤𝑚, as a function of 𝑚, for 𝑚∈{1,2,…,𝑛}.
1b. Find the probability that 𝑀=1.
1c. Find the probability that 𝑀=𝑚 for 𝑚∈{2,3,…𝑛}.
1d. For 𝑛=2, find 𝐄[𝑀] and 𝖵𝖺𝗋(𝑀) as a function of 𝑘.
𝐄[𝑀]=
Var [M] =

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

As 𝑘→∞, 𝐄[𝑀] >

Answer 1

𝑛.