CHAPTER 2 - Probability Distributions

Abstract

This chapter discusses the univariate distribution functions, their moments, and their transforms. Different random variables can have the same distribution functions. If X is a random variable, then its distribution function FX has the following properties: (1) FX is nondecreasing, (2) limx → ∞FX (x) = 1 and limx → –∞, FX (x) = 0, and (3) FX is continuous from the right. From the real analysis or advanced calculus, a function of bounded variation, which is what FX(x) is, has, at most, a countable number of discontinuities, and all discontinuities of such functions are jumps. The normal or Gaussian distribution is the most important of all distribution functions. A distribution function F is said to be discrete if there exists a countable sequence of real numbers and a corresponding sequence of positive numbers. The most frequent example of a continuous singular distribution function is the Cantor distribution or uniform distribution over the Cantor ternary set.