Continuous Distributions

Abstract

The basic ideas of previous sections were the notions of a random variable, its probability distribution, expectation, and standard deviation. These ideas will now be extended from discrete distributions to continuous distributions on a line, in a plane, or in higher dimensions. This chapter concerns continuous probability distributions over an interval of real numbers. One example is the normal distribution, seen already as an approximation to various discrete distributions. A simpler example is the uniform distribution on an interval, defined by relative lengths. Another example, the exponential distribution, treated in Section 4.2, is the continuous analog of the geometric distribution. Each of these distributions is defined by a probability density function, like the familiar normal curve associated with the normal distribution. The way a continuous distribution can be specified by such a density function is the subject of Section 4.1. Change of variable for distributions defined by densities is the subject of Section 4.4.