Some inequalities for moments and coefficients of variation of probability densities over the positive real line are obtained by means of simple geometrical relationships. As an illustrative application rigorous bounds are obtained for the ratio of weight average to number average molecular weight for a large class of distributions of macromolecules, giving a more precise characterization of this empirical measure of heterogeneity.

1. 1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all .

2. 2. Let H be the Hardy–Littlewood maximal function HY(x) = E(Y – X | Y > x). Then HY(Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y.

3. 3. Let X1X2 · ·· Xn be random variables with given marginal distributions, let I1,I2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij, and delay times Xi.) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the ‘bottleneck probability' of each path.

We analyze here the optimal premium rates for an insurance company for the case where the claims process is a stochastic process with unknown parameters. The beliefs of the insurance company about these parameters are represented by some prior distribution function, and the premium rates are determined on the basis of this distribution. The insurer learns about the unknown parameters from the past behavior of the claims process, and revises the prior distribution whenever new information is obtained (i.e., a claims stream is observed).

The learning process introduces multi-period dynamic considerations to the problem of the rate determination, since the rates determined at any period affect the claims process, and thus the information available for future decisions. We show how the optimal rates can be obtained by using some dynamic programming techniques. We also prove that the optimal rates obtained in our case are lower than those determined by an insurance company that does not revise its distribution.

We consider the problem of how to schedule n tasks on to several identical processors to meet the objective of minimising the expected flow-time. The strategy which always serves those tasks whose processing-time distributions have the highest hazard rates is shown to be optimal when these distributions are all exponential.

A series of queues consists of a number of · /M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The ‘output theorem' for the M/M/1 queue is a corollary of this result.