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Analyze distribution of a 3-dice product

DRWQuantitative Trader · Online Assessment
EasyQuantitative TraderOnline AssessmentStatistics & Math

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This question tests the ability to analyze the probability distribution of the product of three independent discrete random variables and apply these statistical properties to practical risk management.

The Problem

You are rolling three fair six-sided dice (each numbered 1 through 6). Let X, Y, and Z represent the outcomes of the three dice. Let the product P be defined as P = X * Y * Z.

Part 1: Distribution Analysis

  1. Calculate the expected value E[P].
  2. Determine the mode(s) of the distribution of P.
  3. Identify the probability that P is a prime number.

Part 2: Market Making Application

Suppose you are running a book where you offer to buy or sell a derivative whose payoff is exactly P. Based on your distribution analysis:

  1. If you must quote a bid-ask spread that is centered on the median of P, what price do you quote? (Note: The median is the value such that P(P = 0.5 and P(P >= median) >= 0.5).
  2. If you sell the derivative at a fixed price equal to the expected value E[P], how does the skewness of the distribution of P affect your risk of 'picking off' (i.e., the likelihood of the outcome being significantly higher than your sale price)?

Constraints

  • Each die is independent and fair.
  • Assume you are calculating exact probabilities using the sample space of 6^3 = 216 equally likely outcomes.

Hints

Hint 1

Since the dice are independent, use the property E[X * Y * Z] = E[X] * E[Y] * E[Z] to simplify the calculation of the expected value.

Hint 2

List the possible prime products; since P = X*Y*Z, P can only be prime if two of the dice are 1 and the third is a prime number.

Hint 3

To find the median, consider sorting all 216 possible outcomes in ascending order and identifying the 108th and 109th values.

Follow-up questions

  • How would the distribution of P change if the dice were weighted such that the face '6' appeared twice as often as any other face?
  • If you were to sample the product of three dice 1,000 times, how would you estimate the standard error of the mean of your observations?
  • In a market-making context, if you observe the distribution is highly right-skewed, how should you adjust your 'fair value' quote relative to the mean to protect against tail risk?