Two Sigma 的 Quant Researcher（简称 QR）岗位，一直被称为“智商天花板”的岗位之一。它的筛选过程极其严格，我最近刚带学员做完一套 Two Sigma Quant Researcher OA ，这里详细复盘一下我遇到的真题和思考过程。整体感觉题目难度不低，考察的是概率推理、期望计算、数据结构设计和几何直觉，综合性非常强。
Part 1: Review of Online Assessment Questions
Problem 1: Probability Decision Problem
Question:
You start with $1. In each round, you can either stop and take your current amount, or continue.
If you continue, with probability p Your money doubles, and with probability (1 – p) You lose everything.
When should you continue?
Solution idea.
At its core, it compares the expected return of "continuing to play" with the certainty of "stopping".
Assume that the current funding is x:
- If stop → gain = x
- If continue → Expected return = 2x p + 0 (1 - p) = 2px
Therefore, when 2p > 1 (i.e. p > 0.5) When the expected gain from continuing the game is greater than stopping.
Conclusion:
- If p > 0.5, one can play all the way from the expectation point of view;
- In practice, however, if one plays infinitely, the ultimate probability of bankruptcy is still 100%, which begs the question of risk preferences and utility functions.
- If p ≤ 0.5, none of the first rounds should start.
This question is a classic, and Two Sigma just loves the way it examines both mathematical reasoning and risk awareness.
Problem 2: Expected Value of a Sequence
Question:
You have a biased coin with probability p You keep tossing until the pattern "HT" appears.
What's the expected number of tosses?
Solution idea.
This question is a classic question in the state expectation method. We define:
- E = the expected number of coin tosses from the initial state to the occurrence of an HT
- E_H = expected number of HTs after an H has been rolled
The state transfer is as follows:
from the initial state:
E = (1 - p)(1 + E) + p (1 + E_H)
The simplification is: E = 1/p + E_H
from the "Already H" state:
E_H = (1 - p) × 1 + p (1 + E_H)
The simplification is: E_H = 1/(1 - p)
Substitutions can be made:
E = 1/p + 1/(1 - p)
The result is beautiful, and in the "small talk" style that Two Sigma has always favored.
Problem 3: Median in a Dynamic Data Stream
Question:
Design a data structure that supports.
addNum(num)- add a numberfindMedian()- return the median of all elements
要求效率尽可能高。
Solution idea.
这题属于标准的算法题，但在 QR OA 里出现，意味着考察的不只是代码实现，还有逻辑思路。
The optimal solution is to use Two piles.:
- max_heap Store the smaller half of the number (top of heap is the larger value)
- min_heap Store the larger half of the number (top of the pile is the smaller value)
Operational Logic:
- 插入时先放入合适的堆，然后平衡两边大小，使两堆元素数之差不超过 1。
- 查找中位数时：
- If two piles are equal in size → return the average of the tops of the two piles
- If unequal → return the top of the heap with more elements
Complexity.
addNum: O(log n)findMedian: O(1)
This question also examines your sensitivity to "online processing of streaming data", which is a very common type of thinking question in QR interview.
Problem 4: Geometric Probability
Question:
A stick of length 1 is broken at two random points, forming three pieces.
What's the probability that these three pieces can form a triangle?
Solution idea.
Let the two breakpoint locations be x and y (assuming x < y), then the length of the three segments is:
a = x, b = y - x, c = 1 - y.
To form a triangle, it must be satisfied:
a + b > c, a + c > b, b + c > a.
By integration (or the geometric area method), the final result is 1/4.
This question is very much in the style of Two Sigma - combining mathematical intuition and probabilistic geometry to examine logical rigor.
Technical interview stage review (45min/round, 2-3 rounds)
After OA, went into technical interview, purely chatty: probability + modeling based, no whiteboard code. Interviewer (PhD mostly) will dig deeper:
- "How is a decision tree constructed for P dynamic updates (e.g., Bayesian posterior)?" (A: POMDP, value iteration.)
- “HT变TH或’HH’，状态转移变吗？”（对称性分析，重叠罚。）
- "Median plus delete operation?" (Tracking counts with HashMap, double heap adjustment.)
- "Non-uniformity of breakpoints in trigonometric probability?" (Dirichlet distribution, integral recalculus.)
Tips: don't just report answers, draw state diagrams/geometry as you go. Emphasize trade-offs (e.g., exact vs. approximate). Behavioral facets interspersed: talk about probability applications in projects (e.g., A/B testing).
Summarize
Two Sigma's QR OA can be described as a concentration of Math Intuition + Programming Logic + Rigor.
If you are preparing for a similar quantitative post test, more practice is highly recommended:
- probability and expectation type questions
- State Transfer + Equation Modeling
- Dynamic data structure maintenance issues
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