It is well known that Poisson binomial distribution can be approximated by Poisson distribution. In this paper, we give a nonuniform bound of this approximation by using Stein-Chen method.

Here is the gist of the Eisenbud–Sturmfels paper Binomial ideals, Duke Math. J. 84 (1996), 1–45. The main results are that the associated primes, the primary components, and the radical of a binomial ideal in a polynomial ring are binomial if the base ring is algebraically closed. Throughout, R = k[X1, . . . , Xn], where k is a field and X1, . . . , Xn are variables over k. A monomial is an ele...

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

Tusnady’s inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian Bridge. We present an elementary proof of a result that sharpens the Tusnady inequality, modulo constants. Our method uses the beta integral representation of binomial tails, simple Taylor expansion, and some novel bounds for the ratios of normal tail probabilities. Ke...

We investigate the KNO scaling function of the modified negative binomial distribution (MNBD), because this MNBD can explain the oscillating behaviors of the cumulant moment observed in e+e− annihilations and in hadronic collisions. By using a straightforward method and the Poisson transform we derive the KNO scaling function from the MNBD. The KNO form of experimental data in e+e− collisions a...

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer k ≥ 3, the binomial k-uniform random hypergraph H n,p contains N := (1 − o(1)) ( n−1 k−1 ) p edge-disjoint perfect matchings, provided p ≥ log C n nk−1 , where C := C(k) is an integer dependin...

For >40 years, geneticists and science historians have appealed to the tetrad-pollen model as an explanation of the bias toward expectation in Mendel's data, albeit without experimental support. Our experiments demonstrate that pollen sampling during self-pollination in pea conforms to the binomial distribution with no evidence of a tetrad-pollen effect.