Order Statistics
Work with the sorted values: minimum, maximum and the -th smallest.
What is it?
Order statistics are the sorted sample. Their distributions follow from simple CDF arguments.
Why is it useful?
Expected max/min of dice or uniforms and "best of" questions appear constantly in quant interviews.
When should you think of it?
Think of it whenever the question involves the largest, smallest or -th value among several draws.
Think of this method when…
- "Expected maximum", "expected minimum", "highest roll"
- The -th smallest or the range
- i.i.d. draws, often uniform
- Auction or best-of- settings
Core idea
exactly when all draws are ; exactly when all are .
Step-by-step playbook
- 1Write the CDF of the max or survival function of the min.
- 2Differentiate for a density, or use the tail-sum formula for the mean.
- 3For uniforms, use directly.
Worked example
Worked example
Three independent variables: what is the expected maximum?
- 1
, so the density is .
- 2
Integrate.
- Answer:
Common mistakes
- Treating the max as the sum of CDFs rather than their product.
- Using the continuous formula for discrete dice.
Would you use this method?
What is the expected minimum of two fair dice?
Practice this method
38 QuantSolve questions are tagged with this method.
Practice this method