Chapter 6 - Probability Distributions of Continuous Variables: For this chapter the reader needs a good knowledge of integral calculus and the material in sections 2.1, 2.2, 5.1, and 5.2

Abstract

This chapter discusses the probability distributions of continuous variables. If a variable is continuous, between any two possible values of the variable are an infinite number of other possible values, even though we cannot distinguish some of them from one another in practice. It is therefore, not possible to count the number of the possible values of a continuous variable. In this situation, calculus provides the logical means of finding probabilities. The chapter explains probability from the probability density function. The basic relationship is well described with a simple illustration. The mathematical expectation or expected value of a discrete random variable is a mean result for an infinitely large number of trials, so it is a mean value that would be approximated by a large but finite number of trials. This also holds true for a continuous random variable. For a discrete random variable, the expected value is found by adding up the product of each possible outcome with its probability. The normal distribution is the continuous distribution, which is by far the most used by engineers. Some are based on the normal distribution, and the corresponding tests assume that the underlying population is at least approximately normally distributed. The other continuous distributions, which should be mentioned are the uniform distribution, the exponential distribution, the Weibull distribution, the beta distribution, and the gamma distribution. Others are important in various specialized applications.