Theory of ϕ-Jensen variance and its applications in higher education

Abstract

This paper introduces the theory of φ-Jensen variance. Our main motivation is to extend the connotation of the analysis of variance and facilitate its applications in probability, statistics and higher education. To this end, we first introduce the relevant concepts and properties of the interval function. Next, we study several characteristics of the log-concave function and prove an interesting quasi-log concavity conjecture. Next, we introduce the theory of φ-Jensen variance and study the monotonicity of the interval function JVar φ ϕ(X [a,b] ) by means of the log concavity.Finally, we demonstrate the applications of our results in higher education, show that the hierarchical teaching model is 'normally' better than the traditional teaching model under the appropriate hypotheses, and study the monotonicity of the interval function Var A (X [a,b] ). MSC: 26D15; 62J10

Sharp upper bound involving circuit layout system

“…Indeed, the proof of Theorem 1.5 is both interesting and difficult. Some techniques related to the proof of Theorem 1.5 can also be found in the references [1]- [3] cited in this paper.…”

A Brunn-Minkowski-type inequality involving γ -mean variance and its applications

“…In [16], the authors generalized the traditional covariance and the variance of random variables, and defined φ-covariance, φ-variance, φ-Jensen variance, φ-Jensen covariance, integral variance, and γ-order variance, as well as they studied the relationships among these variances. Moreover, they dealt with a quasi-log concavity conjecture and the monotonicity of the interval function JVar φ ϕ X [a,b] .…”

“…In [14,16], the authors extended the classic variance Varϕ of the random variable ϕ : Ω → (0, ∞) and defined the γ-order variance as follows:…”

“…Eϕ Ω pϕ, Varϕ Eϕ 2 − (Eϕ) 2 and Varϕ Varϕ are the mathematical expectation, variance and the mean variance of the random variable ϕ(X), respectively, where the function ϕ : Ω → R is continuous (see [14][15][16]). In [15], the authors studied the convergence of the following generalized integral:…”

“…where C(Ω) is a linear space in the real number field R, Cov(f, g) is the covariance of the random variables f (X) and g(X) (see [16]). Then f, g is a quasi-inner product of the functions f and g, which satisfies the following conditions:…”