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Approximate a Rare-Event Binomial Probability

Virtu FinancialQuantitative Researcher · Onsite
HardQuantitative ResearcherOnsiteStatistics & Math

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This problem tests the application of the Poisson approximation to binomial distributions for rare events and the evaluation of the error bounds associated with this approximation.

The Problem

Suppose you are analyzing the frequency of a rare system error in a high-frequency trading engine. The engine processes n = 10,000 trades per second, and each trade has a probability p = 0.0005 of triggering a specific latency alert.

  1. Calculate the exact probability of observing more than 5 latency alerts in a single second using the binomial distribution.
  2. Approximate this probability using the Poisson distribution. Define the appropriate parameter lambda and compute the result using the complement rule: P(X > 5) = 1 - P(X <= 5).
  3. Discuss the theoretical conditions under which the Poisson approximation for a binomial distribution is considered reliable.

Constraints and Considerations

  • Use lambda = n * p.
  • You may express the result using the summation of the Poisson probability mass function: P(X = k) = (lambda^k * e^-lambda) / k!.
  • Explain how the approximation changes if n were to increase to 1,000,000 while keeping the expected value lambda constant.

Hints

Hint 1

Recall that the Poisson distribution serves as the limit of the binomial distribution as n approaches infinity and p approaches 0, while n*p remains constant.

Hint 2

For the complement calculation, compute the sum of Poisson probabilities for k = 0, 1, 2, 3, 4, and 5, then subtract this sum from 1.

Hint 3

Consider the Le Cam's Theorem, which provides an upper bound on the total variation distance between the binomial and Poisson distributions.

Follow-up questions

  • How would the accuracy of your approximation change if the probability p was significantly larger, such as p = 0.1?
  • If the alerts were not independent, but instead occurred in clusters, would the Poisson distribution still be an appropriate model? Why or why not?
  • How does the variance of the binomial distribution compare to the variance of the Poisson approximation, and what does this imply about the quality of the fit?