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Approximate a Binomial Upper Tail with a Normal Distribution

Virtu FinancialQuantitative Researcher · Onsite
HardQuantitative ResearcherOnsiteStatistics & Math

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This question tests the application of the Normal approximation to the Binomial distribution, the use of continuity correction, and the ability to evaluate the error bounds of this approximation.

Problem Statement

Consider a fair coin that is flipped n = 100 times. Let X be a random variable representing the total number of heads observed, where X follows the Binomial distribution B(n, p) with p = 0.5.

Your task is to calculate the probability P(X > 60) using a Normal approximation with continuity correction.

  1. Define the parameters (mean mu and standard deviation sigma) for the approximating Normal distribution.
  2. Apply the continuity correction to estimate P(X > 60) using the Normal cumulative distribution function (CDF). For this calculation, assume the standard Normal CDF table values: Phi(1.96) is approximately 0.975, Phi(2.0) is approximately 0.9772, and Phi(2.04) is approximately 0.9793.
  3. Discuss the quality of this approximation. Given that the exact probability of P(X > 60) is approximately 0.0284, explain why the approximation error occurs.

Constraints

  • Assume the standard Normal distribution Z ~ N(0, 1).
  • Express the final estimate for P(X > 60) as a decimal.
  • Justify the use of the continuity correction factor (0.5) in this context.

Hints

Hint 1

Recall that for a Binomial distribution, the mean is np and the variance is np(1-p).

Hint 2

To approximate a discrete P(X > k) with a continuous distribution, shift the boundary k to k + 0.5 to include the entirety of the bar for k in the discrete histogram.

Hint 3

The quality of the Normal approximation depends on the skewness of the Binomial distribution, which decreases as n increases and p approaches 0.5.

Follow-up questions

  • How would the accuracy of your approximation change if p were 0.05 instead of 0.5?
  • Under what condition of n and p is the Normal approximation considered statistically 'sufficient' for binomial tails?
  • If you were asked to approximate the tail for a much larger n, such as n = 10,000, would the continuity correction remain significant?