[Paper Review] A Link between Binomial Parameters and Means of Bounded Random Variables

Xinjia Chen|ArXiv.org|Feb 27, 2008

Advanced Statistical Methods and Models2 references3 citations

TL;DR

This paper establishes a fundamental identity linking the expected value of a bounded random variable to binomial probabilities via a uniform auxiliary variable. By expressing the mean of a bounded random variable as a linear combination of binomial parameters, the method enables confidence interval construction and sample size determination using well-established binomial inference techniques.

ABSTRACT

In this paper, we establish a fundamental connection between binomial parameters and means of bounded random variables. Such connection finds applications in statistical inference of means of bounded variables.

Motivation & Objective

• To establish a theoretical connection between the mean of a bounded random variable and binomial probabilities.
• To enable statistical inference for means of bounded random variables by reducing the problem to binomial parameter estimation.
• To provide a framework for constructing confidence intervals and solving sample size problems using binomial distribution theory.
• To generalize the link through flexible choices of auxiliary variables and thresholds, including special cases like Corollary 2.
• To support practical applications in engineering and science where bounded random variables are common, such as in performance evaluation and reliability analysis.

Proposed method

• Derives a general identity expressing the expectation of a bounded random variable $X \in [0,c]$ as a function of the expectation of another bounded variable $Y$ and three binomial probabilities.
• Introduces an auxiliary uniform random variable $U \sim \text{Uniform}[0,1]$ independent of $X$ and $Y$ to define threshold events.
• Defines three Bernoulli random variables based on stochastic inequalities involving $X$, $Y$, $U$, and parameters $\delta$, $c$, $a$, $b$, and $c_2$.
• Uses the linearity of expectation and decomposition of the joint distribution to derive the identity through four lemmas on integral representations.
• Applies the Clopper-Pearson method and Bonferroni’s inequality to construct confidence intervals for the binomial parameters.
• Simplifies the general identity to Corollary 2, where $\mathbb{E}[X] = c \cdot \Pr\{X \geq cU\}$, enabling direct scaling of binomial confidence intervals.

Experimental results

Research questions

• RQ1 Can the mean of a bounded random variable be expressed as a linear combination of binomial probabilities?
• RQ2 How can statistical inference for the mean of a bounded random variable be reduced to inference on binomial parameters?
• RQ3 What is the role of an auxiliary uniform random variable in linking bounded means to binomial events?
• RQ4 How can confidence intervals for the mean of a bounded random variable be constructed using binomial inference techniques?
• RQ5 What sample size is required to estimate the mean of a bounded random variable within a given error margin with specified confidence?

Key findings

• The paper establishes a fundamental identity: $\mathbb{E}[X] = \mathbb{E}[Y] + \delta \Pr\{X \geq Y + \delta U\} + |c - a - \delta| \Pr\{X > Y + \delta + |c - a - \delta|U\} - b \Pr\{X < bU < Y\}$, valid for any $\delta$.
• Corollary 1 provides a simplified version with constants $c_1$ and $c_2$, expressing $\mathbb{E}[X]$ as a linear combination of three binomial probabilities.
• Corollary 2 shows that when $c_1 = 0$ and $c_2 = c$, $\mathbb{E}[X] = c \cdot \Pr\{X \geq cU\}$, offering a direct link between the mean and a single binomial probability.
• The method enables construction of confidence intervals for $\mathbb{E}[X]$ by estimating the three binomial parameters using sample proportions and applying Bonferroni’s inequality.
• The approach allows solving sample size problems by scaling the required sample size for binomial proportion estimation to achieve a desired margin of error for the bounded mean.
• The theoretical framework is supported by four lemmas that rigorously decompose the expectation into components involving integrals and probabilities, validated through case analysis on parameter signs.