Probability: A Lively Introduction

In many practical applications of probability, physical situations are better described by random variables that can take on a continuum of possible values rather than a discrete number of values. Examples are the decay time of a radioactive particle, the time until the occurrence of the next earthquake in a certain region, the lifetime of a battery, the annual rainfall in London, electricity consumption in kilowatt hours, and so on. These examples make clear what the fundamental difference is between discrete random variables taking on a discrete number of values and continuous random variables taking on a continuum of values. Whereas a discrete random variable associates positive probabilities with its individual values, any individual value has probability zero for a continuous random variable. It is only meaningful to speak of the probability of a continuous random variable taking on a value in some interval. Taking the lifetime of a battery as an example, it will be intuitively clear that the probability of this lifetime taking on a specific value becomes zero when a finer and finer unit of time is used. If you can measure the heights of people with infinite precision, the height of a randomly chosen person is a continuous random variable. In reality, heights cannot be measured with infinite precision, but the mathematical analysis of the distribution of people's heights is greatly simplified when using a mathematical model in which the height of a randomly chosen person is modeled as a continuous random variable. Integral calculus is required to formulate the continuous analogue of a probability mass function of a discrete random variable.

The first purpose of this chapter is to familiarize you with the concept of the probability density of a continuous random variable. This is always a difficult concept for the beginning student. However, integral calculus enables us to give an enlightening interpretation of probability density. The second purpose of this chapter is to introduce you to important probability densities such as the uniform, exponential, gamma, Weibull, beta, normal, and lognormal densities among others. In particular, the exponential and normal distributions are treated in depth. Many practical phenomena can be modeled by these distributions, which are of fundamental importance. Many examples are given to illustrate this.