Order Statistics Maximum Worked Example
This example teaches the cleanest Exam P order-statistics move: rewrite a statement about the sample maximum as an all-observations event, then use independence.
- Role
- Worked Example
- Level
- Core
- Time
- Reference
- Freshness
- Stable
Problem
Let X1, X2, X3, and X4 be iid Uniform(0,1). Find the probability that the sample maximum is less than 0.7. Then find the probability that the sample maximum exceeds 0.9.
What This Example Is Testing
This problem checks whether you see through the order-statistics wording. A maximum event is really an event about all observations being below a threshold, and its complement is an event about at least one observation crossing the threshold.
Step-By-Step Solution
Let M=X(4) be the maximum. The event M < 0.7 means every observation is below 0.7. Because the Xi are iid Uniform(0,1), each one has probability 0.7 of being below 0.7.
So P(M < 0.7)=0.7^4=0.2401.
For the second part, M > 0.9 means at least one observation exceeds 0.9. The complement is that all observations are at most 0.9, which has probability 0.9^4=0.6561.
Therefore P(M > 0.9)=1-0.9^4=1-0.6561=0.3439.
Final Answer
The probability that the maximum is less than 0.7 is 0.2401. The probability that the maximum exceeds 0.9 is 0.3439.
Common Wrong Answer
A common wrong answer is to use 4 x 0.7 or 4 x 0.1, as if the events were disjoint. They are not. The correct structure uses independence across all four observations and complement logic for the second part.