0495382175_ch04 - 38217_04_c4_p130-183.qxd 12/16/06 16:08...

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130 Continuous Random Variables and Probability Distributions 4 INTRODUCTION Chapter 3 concentrated on the development of probability distributions for dis- crete random variables. In this chapter, we study the second general type of random variable that arises in many applied problems. Sections 4.1 and 4.2 present the basic definitions and properties of continuous random variables and their probability distributions. In Section 4.3, we study in detail the normal ran- dom variable and distribution, unquestionably the most important and useful in probability and statistics. Sections 4.4 and 4.5 discuss some other continuous distributions that are often used in applied work. In Section 4.6, we introduce a method for assessing whether given sample data is consistent with a speci- fied distribution.
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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, and (2) P ( X 5 c ) 5 0 for any number c that is a possible value of X. If in the study of the ecology of a lake, we make depth measurements at randomly chosen locations, then X 5 the depth at such a location is a continuous rv. Here A is the minimum depth in the region being sampled, and B is the maximum depth. n If a chemical compound is randomly selected and its pH X is determined, then X is a continuous rv because any pH value between 0 and 14 is possible. If more is known about the compound selected for analysis, then the set of possible values might be a subinterval of [0, 14], such as 5.5 # x # 6.5, but X would still be continuous. n Let X represent the amount of time a randomly selected customer spends waiting for a haircut before his/her haircut commences. Your first thought might be that X is a continuous random variable, since a measurement is required to determine its value. However, there are customers lucky enough to have no wait whatsoever before climb- ing into the barber’s chair. So it must be the case that P ( X 5 0) . 0. Conditional on no chairs being empty, though, the waiting time will be continuous since X could then assume any value between some minimum possible time A and a maximum possible time B . This random variable is neither purely discrete nor purely continu- ous but instead is a mixture of the two types. n One might argue that although in principle variables such as height, weight, and temperature are continuous, in practice the limitations of our measuring instru- ments restrict us to a discrete (though sometimes very finely subdivided) world.
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This note was uploaded on 09/27/2010 for the course STAT 104157 taught by Professor Jhonbran during the Spring '09 term at California Coast University.

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0495382175_ch04 - 38217_04_c4_p130-183.qxd 12/16/06 16:08...

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