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

Let X be the number of successes in n independent dichotomous trials.Among all sets of success probabilities (p 1, ., pn ) with a given average pbar, X has maximal variance when p1=p2=...=pn=pbar .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.