scribed in the next section. Probability distributions Continuous distributions At the beginning of this chapter, I discussed the possibility that the events form a con- tinuous and infinite suite of real numbers, which, in the physical world, represent the unbounded set of measurements of a physical quantity. The likelihood of the measure- ment then corresponds mathematically to a continuous probability distribution. Like discrete PDF, a continuous PDF p(x) must satisfy the property 0<p(x)<1 (2.10) for all values X belonging to the event space Lxmin, Xmax_ within which the events are defined. By definition, the sum of all probabilities must be equal to unity. With a continuous variable, such a sum is an integral, i.e., x=xmax (2.11) Consistent with the above definition, the PDF mean or expected value <x> and the mean square are defined as follows: (2.12) (2.13) with the variance being o2 = <x2>-<x>2, just as in the discrete-PDF case. Note that for any continuous PDF one can replace the interval [Xmin, Xmax by [-00,+00], with the convention that p(x) = 0 for x [Xmin, Xmax]. Uniform, exponential, and Gaussian (normal) distributions In the following, we shall consider three basic types of continuous distribution: uniform, exponential, and Gaussian (normal). The most elementary type of continuous PDF is the uniform distribution. It gives a good pretext to open up our integral-calculus toolbox. Such a distribution is defined as p(x) = const. over a certain interval [Xmin, Xmax] and p(x) = 0 elsewhere. The constant is found by applying Eq. (2.11), i.e., x=xmax x=xmin (2.14) where = Xmax - Xmin is the sample-space width, which gives const. = 1/Ax. Figure 2.5 shows a plot of the uniform distribution thus defined. The figure also illus- trates that the PDF is actually defined over the infinite space [-00, +00], while being 2.4 Uniform, exponential, and Gaussian distributions Figure 2.5 Uniform probability distribution of the continuous variable X, see text for description. identically zero outside the relevant sample int