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Introduction to Continuous Random Variables

Module by: Roberta Bloom. E-mail the author

Based on: Continuous Random Variables: Introduction by Susan Dean, Barbara Illowsky, Ph.D.

Summary: This module serves as an introduction to the Continuous Random Variables chapter in the Elementary Statistics textbook. The original module by S. Dean and B. Illowsky has been revised; concepts removed from the original version of module are discussed in R. Bloom's module Continuous Random Variables: Properties of Continuous Probability Distributions

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, weight, height, and temperature are just a few. Generally, for continuous random variables, the outcomes are measured, rather than counted. The field of reliability depends on a variety of continuous random variables.

Note that the values of discrete and continuous random variables can sometimes 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.

This chapter gives an introduction to continuous random variables and continuous probability distributions. There are many continuous probability distributions. We will be studying continuous distributions for several chapters and will use continuous probability throughout the rest of this course. We will start with the two simplest continuous distributions, the Uniform and the Exponential.

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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