Some Generalized Probability Distributions with Special Reference to the Mineral Industries. Part I. Sampling to n Items Per Sample

摘要

The investigation extends the theory of sample reliability from a sample consisting of a fixed number of items to a sample consisting of a fixed amount of items. Examples of a fixed amount of items per sample are samples consisting of a fixed weight or a fixed volume of unequal size particles. The extension of the theory is confirmed by experimental laboratory sampling using standard procedures. The probability distribution functions (the binomial, hypergeometric, multinomial, generalized hypergeometric, and those called the compound binomial and compound multimomial functions) are developed from fundamentals. Their distributions are generalized to include some measured characteristic of the items. Infinite population moments are derived from moment-generating functions, and finite population moments are developed from corresponding infinite population moments using a newly defined mathematical operation. The first three experimental moments in laboratory sampling to given weights of particles are compared with the theoretical moments of their corresponding compound multinomial distributions. The agreement is striking. The study is divided into five parts. Part 1 deals with probability or random sampling to n items and with all sampling to n items per sample from some one probability distribution. This is the area of sampling covered by well-known theory. Accordingly, the procedures may be different but many of the results are well known. (Author)