3. Continuous Probability Distributions

Abstract

Researchers use random variables to describe the numerical results of experiments. In the continuous setting, the possible numerical values form an interval or a union of intervals. For example, a random variable whose values are the positive real numbers might be used to describe the lifetimes of individuals in a population.This chapter focuses on continuous random variables and their probability distributions. The first two sections give the important definitions and example families of distributions. Section 3 generalizes the ideas to joint distributions. Section 4 outlines the laboratory problems.3.1 DefinitionsRecall that a random variable is a function from the sample space of an experiment to the real numbers and that the random variable X is said to be continuous if its range is an interval or a union of intervals.3.1.1 PDF and CDF for continuous random variablesIf X is a continuous random variable, then the cumulative distribution function (CDF) of X is defined as follows:F(x)=P(X≤x)for all real numbersx.CDFs satisfy the following properties:1. limx→−∞ F(x) = 0 and limx→+∞ F(x) = 1.2. If x1 ≤ x2, then F(x1) ≤ F(x2).3. F(x) is continuous.F(x) represents cumulative probability, with limits 0 and 1 (property 1). Cumulative probability increases with increasing x (property 2). For continuous random variables, the CDF is continuous (property 3).