Devore Probability Statistics Engineering Sciences 8th txtbk

138 CHAPTER 4 Continuous Random Variables and Probability Distributions4.1 Probability Density FunctionsA discrete random variable (rv) is one whose possible values either constitute a finiteset 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 entireinterval of numbers is not discrete.Recall from Chapter 3 that a random variable X is continuous if (1) possiblevalues comprise either a single interval on the number line (for some A , B, anynumber 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.Example 4.1Example 4.2Example 4.3If in the study of the ecology of a lake, we make depth measurements at randomlychosen locations, then X 5 the depth at such a location is a continuous rv. Here A isthe minimum depth in the region being sampled, and B is the maximum depth. ■If a chemical compound is randomly selected and its pH X is determined, then X isa continuous rv because any pH value between 0 and 14 is possible. If more is knownabout the compound selected for analysis, then the set of possible values might be asubinterval of [0, 14], such as 5.5 # x # 6.5, but X would still be continuous. ■Let X represent the amount of time a randomly selected customer spends waiting fora haircut before his/her haircut commences. Your first thought might be that X is acontinuous random variable, since a measurement is required to determine its value.However, there are customers lucky enough to have no wait whatsoever beforeclimbing 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 continuoussince X could then assume any value between some minimum possible time A and amaximum possible time B. This random variable is neither purely discrete nor purelycontinuous but instead is a mixture of the two types.■One might argue that although in principle variables such as height, weight,and temperature are continuous, in practice the limitations of our measuring instrumentsrestrict us to a discrete (though sometimes very finely subdivided) world.However, continuous models often approximate real-world situations very well, andcontinuous mathematics (the calculus) is frequently easier to work with than mathematicsof discrete variables and distributions.Probability Distributions for Continuous VariablesSuppose the variable X of interest is the depth of a lake at a randomly chosen pointon the surface. Let M 5 the maximum depth (in meters), so that any number in theinterval [0, M] is a possible value of X. If we “discretize” X by measuring depth tothe nearest meter, then possible values are nonnegative integers less than or equal toM. The resulting discrete distribution of depth can be pictured using a probability histogram.If we draw the histogram so that the area of the rectangle above any possibleinteger k is the proportion of the lake whose depth is (to the nearest meter) k, then thetotal area of all rectangles is 1. A possible histogram appears in Figure 4.1(a).If depth is measured much more accurately and the same measurement axis asin Figure 4.1(a) is used, each rectangle in the resulting probability histogram is muchnarrower, though the total area of all rectangles is still 1. A possible histogram isCopyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.