Random Variables: The Continuous Case
5.1 Introduction/Purpose of the Chapter
In the previous chapter we discussed the properties of discrete random variables which map events to values in a countable set. In many cases, however, we need to consider variables which take values in an interval. Think about the following experiment: Choose a random point on a segment from the origin to some point A and let be X the abscissa of the chosen point. Then X(Ω) = [0, |A|x], where |A|x is the x-coordinate of the point A and this set is not countable. A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values. Informally, a random variable X is called continuous if its values x form a “continuum,” with P(X = x) = 0 for each x.
5.2 Vignette/Historical Notes
Historically the continuous random variables appeared as approximations of the discrete random variables. The 1756 edition of The Doctrine of Chance contained what is probably de Moivre's most significant contribution to probability, namely the approximation of the binomial distribution with the normal distribution in the case of a large number of trials—which is now known by most probability textbooks as “The First Central Limit Theorem.” Pierre-Simon de Laplace (1749–1827) published Théorie Analytique des Probabilités in 1812. In this book he introduces what is now known as the Laplace transform in applied mathematics, and we will know it as the moment-generating function ...