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For several pairs (P, Q) of classical distributions on N 0 , we show that their stochastic ordering P ≤ st Q can be characterized by their extreme tail ordering equivalent to P ({k * })/Q({k * }) ≥ 1 ≥ lim k→k * P ({k})/Q({k}), with k * and k * denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P ({k * })/Q({k * }) for k * finite. This includes in particular all pairs where P and Q are both binomial (b n1,p1 ≤ st b n2,p2 if and only if n 1 ≤ n 2 and (1 − p 1 ) n1 ≥ (1 − p 2 ) n2 , or p 1 = 0), both negative binomial (b − r1,p1 ≤ st b − r2,p2 if and only if p 1 ≥ p 2 and p r1 1 ≥ p r2 2 ), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new.The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv) and (v)). The statement for hypergeometric distributions is proved via method (i). MSC: primary 60E15

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Patient-indexed norovirus outbreaks generally affect more patients than do staff-indexed outbreaks. Staff appear to be similarly affected by both outbreak index category groups. This study demonstrates the importance of obtaining complete outbreak data, including the index case classification as staff or patient, during norovirus outbreak investigations. Such information may be useful for further targeting prevention measures

The optimal constant over square root of n error bound in the central limit theorem for distribution functions of sums of independent symmetric Bernoulli random variables is 1/ √ 2πn.