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Bernoulli Trials, Poisson Trials, Surprising Variances, and Jensen's Inequality

Jerry Nedelman and Ted Wallenius
The American Statistician
Vol. 40, No. 4 (Nov., 1986), pp. 286-289
DOI: 10.2307/2684605
Stable URL: http://www.jstor.org/stable/2684605
Page Count: 4
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Bernoulli Trials, Poisson Trials, Surprising Variances, and Jensen's Inequality
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Abstract

Let X be the number of successes in n independent dichotomous trials. Among all sets of success probabilities (p1,..., pn) with a given average p̄, X has maximal variance when p1 = p2 = ⋯ = pn, = p̄. Many authors have called this fact surprising. We suggest how to exploit the surprise to lead first-year probability students through a discussion about convex functions, Jensen's inequality, and compound distributions.

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