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A refinement of the binomial distribution using the quantum binomial theorem

Andrew V. Sills

Abstract

$q$-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, $q$-analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the the noncommutative $q$-binomial theorem), where the $q$ is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent.

A refinement of the binomial distribution using the quantum binomial theorem

Abstract

-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, -analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the the noncommutative -binomial theorem), where the is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent.

Paper Structure

This paper contains 12 sections, 7 theorems, 53 equations.

Table of Contents

  1. Background and motivation
  2. Introduction
  3. On the binomial distribution
  4. Interpretation of the exponent of $q$
  5. A refined binomial distribution
  6. The joint distribution of $Y$ and $T$
  7. A summation lemma
  8. Various moments
  9. The conditional distribution of $T$ given $Y$
  10. The conditional distribution of $Y$ given $T$
  11. Example/Application
  12. Conclusion

Key Result

Theorem 2.1

For non-commuting indeterminates $x$ and $y$ and the operator $Q^{\lambda}$ defined in prodtrans,

Theorems & Definitions (16)

  • Theorem 2.1: generalized quantum binomial theorem
  • Corollary 2.2: Quantum binomial theorem
  • Corollary 2.3: classical binomial theorem
  • proof : Proof of Theorem \ref{['gqbt']}
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Lemma 5.1
  • proof
  • ...and 6 more