ABSTRACT
This chapter present convincing reasons for why a solid theoretical probability foundation is needed in the development of statistics. Statisticians know that the subject of statistics is concerned with the development and application of statistical models — descriptions in which there are significant factors affecting what is observed, but which are not accounted for explicitly. Given the formula for some probability function or probability density, e.g., a Poisson probability function, what is a formula for the population mean and for the population standard deviation — the values approached by the sample mean and sample standard deviation respectively, for large sample sizes? It can be shown theoretically that for a Poisson distribution the population standard deviation is the square root of the population mean. Based on Monte Carlo simulations, statisticians can get a good idea of how many observations are needed to accurately estimate the probability of success of an experiment with a specified probability.
