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Analyze distribution of a 3-dice product
Analyze Distribution of a 3-Dice Product
You are rolling three independent fair 10-sided dice. Each die shows an integer in the set {0, 1, ..., 9} with equal probability. Let the outcomes be X, Y, and Z. Define the product P = XYZ.
- Mean: Compute the expected value of P, E[P].
- Median Inference:
- (a) Find the population median of P. This is the smallest value 'm' such that the probability of P being less than or equal to 'm' is at least 0.5 (Pr(P ≤ m) ≥ 0.5).
- (b) Assume you roll the three dice 'n' times independently and compute the sample median, denoted as P̃. Describe how you would construct an approximate 90% confidence interval for the population median using P̃. A rough numerical interval is acceptable if you clearly state the assumed value of 'n'.
- Mode and Skewness:
- (a) What is the mode of P?
- (b) How do the mode and skewness inform your approach to placing or adjusting the confidence interval from part (2b)?
- Market-Making (Interactive): You need to quote a bid and ask price for the population median 'm' of P, with a width of 5 (ask - bid = 5). The interviewer will simulate trades for 4 rounds:
- If the interviewer buys, you sell 1 unit at the ask price (you go short).
- If the interviewer sells, you buy 1 unit at the bid price (you go long). After 4 rounds, the position is settled at the true median 'm'. The profit and loss (P&L) for each unit is (sell price - m) if you sold, or (m - buy price) if you bought.
- (a) Propose an initial bid/ask market [b, a].
- (b) Explain how you would update the bid/ask market after each trade to incorporate the revealed information.
- (c) Show how to compute the resulting P&L given any sequence of 4 buys/sells.
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Analyze distribution of a 3-dice product
Analyze Distribution of a 3-Dice Product
You are rolling three independent fair 10-sided dice. Each die shows an integer in the set {0, 1, ..., 9} with equal probability. Let the outcomes be X, Y, and Z. Define the product P = XYZ.
- Mean: Compute the expected value of P, E[P].
- Median Inference:
- (a) Find the population median of P. This is the smallest value 'm' such that the probability of P being less than or equal to 'm' is at least 0.5 (Pr(P ≤ m) ≥ 0.5).
- (b) Assume you roll the three dice 'n' times independently and compute the sample median, denoted as P̃. Describe how you would construct an approximate 90% confidence interval for the population median using P̃. A rough numerical interval is acceptable if you clearly state the assumed value of 'n'.
- Mode and Skewness:
- (a) What is the mode of P?
- (b) How do the mode and skewness inform your approach to placing or adjusting the confidence interval from part (2b)?
- Market-Making (Interactive): You need to quote a bid and ask price for the population median 'm' of P, with a width of 5 (ask - bid = 5). The interviewer will simulate trades for 4 rounds:
- If the interviewer buys, you sell 1 unit at the ask price (you go short).
- If the interviewer sells, you buy 1 unit at the bid price (you go long). After 4 rounds, the position is settled at the true median 'm'. The profit and loss (P&L) for each unit is (sell price - m) if you sold, or (m - buy price) if you bought.
- (a) Propose an initial bid/ask market [b, a].
- (b) Explain how you would update the bid/ask market after each trade to incorporate the revealed information.
- (c) Show how to compute the resulting P&L given any sequence of 4 buys/sells.