On fractional linear bounds for probability generating functions

The best upper and lower bounds for any probability generating function with mean m and finite variance are derived within the family of fractional linear functions with mean m. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions.

On fractional linear bounds for probability generating functions

The best upper and lower bounds for any probability generating function with meanmand finite variance are derived within the family of fractional linear functions with meanm. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions.

Inequalities with application in economic risk analysis

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

Inequalities with application in economic risk analysis

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

Bounds for the extinction probability of a simple branching process

The extinction probability q of a supercritical simple branching process is well known to be less than unity. Intuitively, it is apparent that when the offspring mean is close to one, so, usually, will q be. This notion is made rigorous, and simple bounds are given for q in terms of the second and third factorial moments, which are applicable when the offspring mean is close to unity. A comparison is made of various upper bounds for q. The note contains some numerical examples.

Some inequalities on variances

Sharp upper and lower bounds for the variance of a non-negative function of a non-negative random variable are obtained under rather weak hypotheses regarding the function. Comparisons between bounds are made and some specific examples are considered.

Attainable bounds for expectations

A simple technique for obtaining bounds in terms of means and variances for the expectations of certain functions of random variables in a given class is examined. The bounds given are sharp in the sense that they are attainable by at least one random variable in the class. This technique is applied to obtain bounds for moment generating functions, the coefficient of skewness and parameters associated with branching processes. In particular an improved lower bound for the Malthusian parameter in an age-dependent branching process is derived.

A bound for bivariate probability of large deviations

Suppose (X 1, Y 1), (X 2, Y 2), …, (Xn , Yn ) are independent random vectors such that a ≦ Xi ≦ b and a ≦ Yi ≦ b, i = 1, 2, …, n. An upper bound which exponentially converges to zero is derived for the probability Pr{Sx – nμ x ≧ nt 1;SY – nμY ≧ nt 2} where Sx = Σ Xi , SY = Σ Yi ,EYi = μY, EXi = μx and t 1 > 0, t2 > 0. The bound is a function of the difference b — a, the correlation between Xi and Yi, μx and μY and t 1 and t 2 .

Polynomial bounds for probability generating functions

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN (·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

Some remarks on probability inequalities for sums of bounded convex random variables

Let X 1, X 2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi , i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi ), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.