Theory of \(\phi\)-Jensen variance and its applications in higher education. (English) Zbl 1334.62135
Summary: This paper introduces the theory of \(\phi\)-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 \(\phi\)-Jensen variance and study the monotonicity of the interval function \(\operatorname{JVar}_{\phi}\varphi ( {{X _{ [ {a,b} ] }}} )\) 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 \(\operatorname{Var} \mathcal{A} (X _{[{a,b}]})\).
MSC:
| 62J10 | Analysis of variance and covariance (ANOVA) |
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
hierarchical teaching model; truncated random variable; interval function; \(\phi\)-Jensen variance; \(k\)-normal distributionReferences:
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