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Continuous Random Variables: Introduction

Module by: Dr. Barbara Illowsky, Susan Dean

Summary: This module serves as an introduction to the Continuous Random Variables chapter in the Elementary Statistics textbook.

Student Learning Objectives

By the end of this chapter, the student should be able to:

  • Recognize and understand continuous probability distribution functions in general.
  • Recognize the uniform probability distribution and apply it appropriately.
  • Recognize the exponential probability distribution and apply it appropriately.

Introduction

Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a long distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables.

This chapter gives an introduction to continuous random variables and the many continuous distributions. We will be studying these continuous distributions for several chapters.

The characteristics of continuous random variables are:

  • The outcomes are measured, not counted.
  • Geometrically, the probability of an outcome is equal to an area under the density curve, fxfx.
  • Each individual value has zero probability of occurring. Instead we find the probability that the value is between two endpoints.

We will start with the two simplest continuous distributions, the Uniform and the Exponential.

NOTE:

The values of discrete and continuous random variables can be ambiguous. For example, if XX is equal to the number of miles (to the nearest mile) you drive to work then XX is a discrete random variable. You count the miles. If XX is the distance you drive to work, then you measure values of XX and XX is a continuous random variable. How the random variable is defined is very important.

Glossary

Uniform Distribution:
Continuous random variable (RV) that appears to have equally likely outcomes over the domain, a<x<ba<x<b size 12{a<x<b} {}. Often referred as Rectangular distribution because graph of its pdf has form of rectangle. Notation: X~U(a,b)X~U(a,b) size 12{X "~" U \( a,b \) } {}. The mean is μ=a+b2μ=a+b2 size 12{μ= { {a+b} over {2} } } {}, and the variance is σ2=(b−a)212σ2=(b−a)212 size 12{s rSup { size 8{2} } = { { \( b-a \) rSup { size 8{2} } } over {"12"} } } {}, the probability density function is f(x)=1b−a,a≤X≤bf(x)=1b−a,a≤X≤b size 12{f \( x \) = { {1} over {b-a} } ," "a <= X <= b} {}, and cumulative distribution is P(X≤x)=x−ab−aP(X≤x)=x−ab−a size 12{P \( X <= x \) = { {x-a} over {b-a} } } {}.
Exponential Distribution:
Continuous random variable (RV) that appears when we are interested in intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital. Notation: X ~ Exp(m)X ~ Exp(m) size 12{X " ~ " ital "Exp" \( m \) } {}; the mean is μ=1mμ=1m size 12{μ= { {1} over {m} } } {}, and the variance is σ 2 = 1 m 2 σ 2 = 1 m 2 , the probability density function is f(x)=me−mx,f(x)=me−mx, size 12{f \( x \) = ital "me" rSup { size 8{- ital "mx"} } ," "} {} x ≥ 0 x ≥ 0 and cumulative distribution is P(X≤x)=1−e−mxP(X≤x)=1−e−mx size 12{P \( X <= x \) =1-e rSup { size 8{- ital "mx"} } } {}.

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