Statistical Inference: A Short Course

Appendix 7.C Randomness, Risk, and Uncertainty¹

7.C.1 Introduction to Randomness

What constitutes “randomness?” Basically, the term random is synonymous with “unpredictable.” For example, if I flip a fair coin repeatedly, then the outcome on any flip (heads or tails) is random—it cannot be accurately predicted from trial to trial with any regularity (although there is a 50/50 chance of correctly guessing the outcome on any given toss of the coin).

This is in contrast to a situation in which one starts with a particular rule for generating outcomes with a fixed initial condition and then proceeds to produce a string of outcomes (numbers). Even though a fairly complicated pattern may emerge from the process and the pattern is indistinguishable from one produced by a purely random process (e.g., a coin toss), the bottom line is that this is not a truly random process. The process is purely deterministic and orderly since, from the known rule, the outcome at any future step is completely predictable. The values generated in this fashion are called pseudo-random numbers and will be discussed in somewhat greater detail below.

What if we observe a string of outcomes (ostensibly random) but we do not have the rule that produced it? If we can encode the pattern of outcomes and ultimately find the rule, then here too we can predict the system's behavior over time and thus pure randomness is not present. Note that in order to predict a future outcome, all steps leading to that outcome have ...