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Analyze distribution of a 3-dice product

Q: Analyze distribution of a 3-dice product

This question evaluates probability and statistical inference skills including computation of expectation, properties of discrete order statistics (population median and mode), sampling variability and confidence-interval construction, and practical market-making reasoning about bid–ask quoting and P&L; it is categorized in Statistics & Math with emphasis on probability, order statistics, estimation, and basic market microstructure. It is commonly asked to assess quantitative reasoning about discrete distributional behavior, robustness to skew and sampling effects, and the ability to apply conceptual distributional understanding to practical estimation and interactive pricing tasks, combining conceptual understanding with practical application.

You roll three independent fair 10-sided dice. Each die shows an integer in $\{0,1,\dots,9\}$ with equal probability. Let the outcomes be $X,Y,Z$ , and define the product $P = XYZ$ .

Mean : Compute $\mathbb{E}[P]$ .
Median inference :
- (a) Find the population median of $P$ (i.e., the smallest $m$ such that $\Pr(P\le m)\ge 0.5$ ).
- (b) Suppose you roll the three dice $n$ times i.i.d. and compute the sample median $\tilde P$ . Describe how you would construct an approximate 90% confidence interval for the population median using $\tilde P$ . (A rough numerical interval is OK if you make clear what $n$ you are assuming.)
Mode and skew :
- (a) What is the mode of $P$ ?
- (b) How does the mode/skewness inform how you would place or adjust the interval from (2b)?
Market-making (interactive) : You must quote a bid/ask market of width 5 for the (unknown-to-the-interviewer) population median $m$ $m$ of $P$ $P$ : you quote a bid $b$ $b$ and ask $a$ $a$ with $a-b=5$ $a - b = 5$ . The interviewer may trade 1 unit each round:
- If they buy , you sell 1 unit at price $a$ (you become short).
- If they sell , you buy 1 unit at price $b$ (you become long).
After 4 rounds, the position is settled at the true median $m$ $m$ (cash-settled), so P&L for each unit is (sell price $-m$ $- m$ ) if you sold, or ( $m$ $m$ -buy price) if you bought.
- (a) Propose an initial market $[b,a]$ .
- (b) Explain how you would update $[b,a]$ after each trade to incorporate the information revealed.
- (c) Show how to compute the resulting P&L given any sequence of 4 buys/sells.

You roll three independent fair 10-sided dice. Each die shows an integer in $\{0,1,\dots,9\}$ with equal probability. Let the outcomes be $X,Y,Z$ , and define the product $P = XYZ$ .

Mean : Compute $\mathbb{E}[P]$ .
Median inference :
- (a) Find the population median of $P$ (i.e., the smallest $m$ such that $\Pr(P\le m)\ge 0.5$ ).
- (b) Suppose you roll the three dice $n$ times i.i.d. and compute the sample median $\tilde P$ . Describe how you would construct an approximate 90% confidence interval for the population median using $\tilde P$ . (A rough numerical interval is OK if you make clear what $n$ you are assuming.)
Mode and skew :
- (a) What is the mode of $P$ ?
- (b) How does the mode/skewness inform how you would place or adjust the interval from (2b)?
Market-making (interactive) : You must quote a bid/ask market of width 5 for the (unknown-to-the-interviewer) population median $m$ $m$ of $P$ $P$ : you quote a bid $b$ $b$ and ask $a$ $a$ with $a-b=5$ $a - b = 5$ . The interviewer may trade 1 unit each round:
- If they buy , you sell 1 unit at price $a$ (you become short).
- If they sell , you buy 1 unit at price $b$ (you become long).
After 4 rounds, the position is settled at the true median $m$ $m$ (cash-settled), so P&L for each unit is (sell price $-m$ $- m$ ) if you sold, or ( $m$ $m$ -buy price) if you bought.
- (a) Propose an initial market $[b,a]$ .
- (b) Explain how you would update $[b,a]$ after each trade to incorporate the information revealed.
- (c) Show how to compute the resulting P&L given any sequence of 4 buys/sells.

Analyze distribution of a 3-dice product

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Analyze distribution of a 3-dice product

Quick Overview

Solution

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