A Probabilistic Approach to The Moments of Binomial Random Variables and Application

  title={A Probabilistic Approach to The Moments of Binomial Random Variables and Application},
  author={Duy Ngoc Nguyen},
  journal={The American Statistician},
  pages={101 - 103}
  • D. Nguyen
  • Published 4 November 2019
  • Mathematics, Computer Science
  • The American Statistician
Abstract In this paper, we provide a closed form formula for the moments of binomial random variables using a probabilistic approach. As an interesting application, we give a closed form formula for the sum . 

Handy Formulas for Binomial Moments.

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not



Raw and central moments of binomial random variables via Stirling numbers

We consider here the problem of calculating the moments of binomial random variables. It is shown how formulae for both the raw and the central moments of such random variables may be obtained in a

Closed-form Expressions for the Moments of the Binomial Probability Distribution

This work develops closed-form expressions for the raw and central moments of the binomial probability distribution from the moment generating function and discusses an application of these formulae to the analysis of neural associative memory.

A First Course in Probability

1. Combinatorial Analysis. 2. Axioms of Probability. 3. Conditional Probability and Independence. 4. Random Variables. 5. Continuous Random Variables. 6. Jointly Distributed Random Variables. 7.

Probabilistic Proof that .

Let X be the sum of two n-sided dice. For k=2,3,…,n+1 , the probability that X = k is (k−1)/n2 because there are k – 1 ways of adding two positive integers to k, namely 1+(k−1),2+(k−2),…,(k−1)+1 . ...

A Recursive Formula for Moments of a Binomial Distribution

While teaching a course in probability and statistics, one of the authors came across an apparently simple question about the computation of higher order moments of a random variable. The topic of

A Combinatorial Approach to Sums of Integer Powers

n(n + )6(26 + 1), makes this fact plausible even to the beginner and might also spark the conjecture that the resulting polynomial should have degree p + 1. Proofs that validate this or produce a


This course can be used as a preparation for the first (Probability) actuarial exam and the central limit theorem and classical sampling distributions.

A short proof of a sum of powers formula

To simplify, make the change of variable = i+ 1 and count the tuples (a1, a2, . . . , ak, ) such that 1 ≤ am < ≤ n+ 1.

Sums of Powers of Integers

For some of the history of the subject, and for a selection of these articles, we mention [1], [3], [5], [7], [9], [11], [12], [13] and [16], and especially [6], [8] and [10]. Here, we shall take a