Let U1,…,Un be i.i.d. random variables uniformly distributed in [0,1] and let Mn=max1≤i≤nUi .
Find the cdf of Mn , which we denote by G(t) , for t∈[0,1] .
To find the cumulative distribution function (CDF) of Mn, we need to determine the probability that Mn is less than or equal to a given value t, for 0 ≤ t ≤ 1.
Since U1,...,Un are independent and uniformly distributed between 0 and 1, we can find the CDF of Mn by calculating the probability that each of the random variables Ui is less than or equal to t, and then taking the maximum of these probabilities.
Let's break down the steps to find G(t):
Step 1: Find the probability that a single random variable Ui is less than or equal to t.
Since Ui is uniformly distributed between 0 and 1, the probability that Ui is less than or equal to t is simply t, denoted as P(Ui ≤ t) = t.
Step 2: Find the probability that all n random variables U1,...,Un are less than or equal to t.
Since the random variables U1,...,Un are independent, we can multiply the probabilities for each individual Ui to find the probability that all n of them are less than or equal to t.
P(U1 ≤ t, U2 ≤ t, ..., Un ≤ t) = P(U1 ≤ t) * P(U2 ≤ t) * ... * P(Un ≤ t) = t^n
Step 3: Find the probability that Mn is less than or equal to t.
To find the CDF of Mn, we need to find the probability that Mn is less than or equal to t, which is the maximum of the probabilities calculated in Step 2.
G(t) = P(Mn ≤ t) = P(U1 ≤ t, U2 ≤ t, ..., Un ≤ t) = t^n
Therefore, the CDF of Mn, denoted by G(t), is equal to t^n for t ∈ [0,1].