We prove an equality without using expectation explicitly for the variance of a random variable by using identical independent random variables.Chebyshev's order inequality are also introduced to emphasize this probabilistic method.Motivated by two fundamental identities of variance,we show some results on the defects of Jensen's inequalities via variance.Bregman's divergence and the role of expectation are introduced.Some algebraic identities,e.g.Huygens-Leibniz's identity and the second Lagrange's identity,are obtained as consequences of the two fundamental identities of variance.Applications of these algebraic identities in the teaching of mathematical statistics are also presented.Finally we present a discussion of strategies for teaching simple facts and difficult subjects.
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