Breaking Sticks, Tossing Needles, and More: Probability on Continuous Sample Spaces

Abstract

Up to now, we have been working with discrete sample spaces and discrete random variables (see Chapters 1 and 7). There are problems, however, for which a discrete sample space is not appropriate because there are just too many possible outcomes. Suppose, for instance, that I want to choose a number on the interval between 0 and 1 “at random.” Ignoring for a moment exactly what I mean by the term “at random” in this context, we note that the number chosen can be any value on the interval, so there is a continuum of possible values. This continuum has so many numbers in it that they cannot all be counted off using the positive integers; for this reason the sample space of this interval is not discrete but is what is called a continuous sample space.