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Tutorial on large deviations for the binomial distribution.

R Arratia, L Gordon

    Bulletin of Mathematical Biology
    |January 1, 1989
    PubMed
    Summary
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    This study simplifies large deviation theory for binomial distributions. It provides an easy approximation for the probability of achieving a certain fraction of successes in multiple trials.

    Area of Science:

    • Probability Theory
    • Statistical Analysis
    • Applied Mathematics

    Background:

    • Binomial distribution is fundamental in probability.
    • Large deviation theory provides approximations for probabilities of rare events.
    • Accurate probability approximations are crucial in statistical modeling.

    Purpose of the Study:

    • To present a user-friendly formulation of large deviation theory for binomial distributions.
    • To develop an approximation for the probability of k or more successes in n trials.
    • To address scenarios where the success fraction (a) exceeds the individual trial success probability (p).

    Main Methods:

    • Application of large deviation theory principles.
    • Derivation of approximation formulas for binomial probabilities.

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  • Focus on the regime where 0 < p < a < 1.

    • Main Results:

    • An accessible method for approximating tail probabilities of the binomial distribution.
    • The approximation is specifically tailored for the case where the observed success fraction is greater than the underlying success probability.
    • Provides a practical tool for estimating the likelihood of exceeding a certain number of successes.

    Conclusions:

    • The presented approach simplifies the application of large deviation theory for binomial probabilities.
    • Offers a valuable tool for researchers and practitioners dealing with binomial data.
    • Enhances understanding of rare event probabilities in binomial settings.