In this paper, we provide a closed form formula for the moments of binomial random variables using a probabilistic approach. As an interesting application, we give a closed form formula for the sum $1^k+2^k+3^k+\dots +n^k$.

This work reviews the literature on spline local basis methods for non-parametric density estimation. Particular attention is paid to B-spline density estimators which have experienced recent advances in both theory and methodology. These estimators occupy a very interesting space in statistics, which lies aptly at the cross-section of numerous statistical frameworks. New insights, experiments, and analyses are presented to cast the various estimation concepts in a unified context, while parallels and contrasts are drawn to the more familiar contexts of kernel density estimation. Unlike kernel density estimation, the study of local basis estimation is not yet fully mature, and this work also aims to highlight the gaps in existing literature which merit further investigation.