A Link between Binomial Parameters and Means of Bounded Random Variables ∗

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. 1 A Fundamental Identity Statistical inference of means of bounded random variables is frequently encountered in many areas of sciences and engineering. Since there exist rich techniques for the statistical inference of probability of events based on binomial trial models, it is useful to express means of bounded random variables in terms of probability of events. In this regard, we have derived a fundamental identity in the following theorem. Theorem 1 Let X be a bounded random variable such that 0 ≤ X ≤ c. Let Y be a bounded random variable such that a ≤ Y ≤ b. Let U be a random variable with a uniform distribution over [0, 1]. Suppose that U is independent with X and Y . Then, E[X ] = E[Y ] + δPr{X ≥ Y + δU}+ |c− a− δ|Pr{X > Y + δ + |c− a− δ|U} − bPr{X < bU < Y } for any real number δ. In Theorem 1, if we set Y = c1 and δ = c2 − c1 with constants c1 and c2, then we have Corollary 1 Let X be a bounded random variable such that 0 ≤ X ≤ c. Let 0 ≤ c1 ≤ c2 ≤ c. Let U be a random variable with a uniform distribution over [0, 1]. Suppose that X and U are independent. Then, E[X] = c1+(c2−c1) Pr{X ≥ c1+(c2−c1)U}+(c−c2) Pr{X > c2+(c−c2)U}−c1 Pr{X < c1U}. The author is currently with Department of Electrical Engineering, Louisiana State University at Baton Rouge, LA 70803, USA, and Department of Electrical Engineering, Southern University and A&M College, Baton Rouge, LA 70813, USA; Email: [email protected]