UNIVARIATE CONTINUOUS RANDOM VARIABLES
In Chapter 2 we briefly discussed a continuous sample space. Following Definition 3.1.2, we define a continuous random variable as a real-valued function defined over a continuous sample space. Or we can define it in a way analogous to Definition 3.2.1: A continuous random variable is a variable that takes a continuum of values in the real line according to the rule determined by a density function. We need to make this definition more precise, however. The following defines a continuous random variable and a density at the same time.
DEFINITION 3.3.1 If there is a nonnegative function /(x) defined over the whole line such that
(3.3.1) P(x ^ X ^ x2) = 2f(x)dx
for any Xi, x2 satisfying Xj < x2, then X is a continuous random variable and / (x) is called its density function.
We assume that the reader is familiar with the Riemann integral. For a precise definition, see, for example, Apostol (1974). For our discussion it is sufficient for the reader to regard the right-hand side of (3.3.1) simply as the area under f(x) over the interval [x1; x2].
We shall allow x = —and/or x2 = °o. Then, by axiom (2) of probability, we must have /“Xf(x)dx = 1. It follows from Definition 3.3.1 that the probability that a continuous random variable takes any single value is zero, and therefore it does not matter whether < or ^ is used within the probability bracket. In most practical applications, /(x) will be continuous except possibly for a finite number of discontinuities. For such a
function the Riemann integral on the right-hand side of (3.3.1) exists, and therefore / (x) can be a density function.