Note on the Binomial Distribution

A Note on N Estimators for the Binomial Distribution

Consider k success counts from a binomial distribution with unknown N and success probability p. We consider the problem of estimating N. By integrating the likelihood for Nand p over a beta density for p, we obtain the beta-binomial distribution which results in stable and reasonably efficient estimators of N, which compare favorably with and are often better than the estimates introduced by O...

Note on the Convolution of Binomial Coefficients

We prove that, for every integer a, real numbers k and ℓ, and nonnegative integers n, i and j, i+j=n a i + k − ℓ i a j + ℓ j = i+j=n a i + k i a j j , by presenting explicit expressions for its value. We use the identity to generalize a recent result of Chang and Xu, and end the paper by presenting, in explicit form, the ordinary generating function of the sequence 2n+k n ∞ n=0 , where k ∈ R.

Binomial Sampling and the Binomial Distribution

This is a good place to point out that these are only (approximating) , and not full truth. models Truth is not a model. A model, by definition, is an approximation (and not truth). In fact, out of a few million coin flips, one or more coins will remain standing on edge – that is truth. The binomial model does not allow such reality. The idea is more easily understood for the outcomes “dead" or...

A Note on Teaching Binomial Confidence Intervals

A Note on Extended Binomial Coefficients

We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as sums of Hermite polynomials and Bernoulli numbers.