4.1 Probability Density Functions

Contents

• 4.1.1 Probability Distributions for Continuous Variables
• EXERCISES Section 4.1 (1-10)

Introduction

A discrete random variable (rv) is one whose possible values either constitute a finite set or else can be listed in an infinite sequence (a list in which there is a first element, a second element, etc.). A random variable whose set of possible values is an entire interval of numbers is not discrete.

Recall from Chapter 3 that a random variable $X$ is continuous if

1. possible values comprise
   • either a single interval on the number line (for some $A<B$ , any number $x$ between $A$ and $B$ is a possible value)
   • or a union of disjoint intervals
2. $P\left(X=c\right)=0$ for any number $c$ that is a possible value of $X$ .

• EX 4.1 depth of lake
• EX 4.2 pH value of chemical compound
• EX 4.3 waiting time of haircut

One might argue that although in principle variables such as height, weight, and temperature are continuous, in practice the limitations of our measuring instruments restrict us to a discrete (though sometimes very finely subdivided) world. However, continuous models often approximate real-world situations very well, and continuous mathematics (the calculus) is frequently easier to work with than mathematics of discrete variables and distributions.