Section 4.5 Probability and Statistics
Modern science may be characterized by a systematic collection of empirical measurements and the attempt to model laws of nature using mathematical language. The drive to deliver better measurements led to the development of more accurate and more sensitive measurement tools. Nonetheless, at some point it became apparent that measurements may not be perfectly reproducible and any repeated measurement of presumably the exact same phenomena will typically produce variability in the outcomes. On the other hand, scientists also found that there are general laws that govern this variability in repetitions. For example, it was discovered that the average of several independent repeats of the measurement is less variable and more reproducible than each of the single measurements themselves.
Probability was first introduced as a branch of mathematics in the investigation of uncertainty associated with gambling and games of chance. During the early 19th century probability began to be used in order to model variability in measurements. This application of probability turned out to be very successful. Indeed, one of the major achievements of probability was the development of the mathematical theory that explains the phenomena of reduced variability that is observed when averages are used instead of single measurements. In Chapter 7 we discuss the conclusions of this theory.
Statistics study method for inference based on data. Probability serves as the mathematical foundation for the development of statistical theory. In this chapter we introduced the probabilistic concept of a random variable. This concept is key for understanding statistics. In the rest of Part I of this book we discuss the probability theory that is used for statistical inference. Statistical inference itself is discussed in Part II of the book.
