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Unformatted text preview: Ch.3 Random Variables and Their Distributions 1 Introduction In Chapter 1 we briefly introduced the concept of a random variable as the numerical description of the outcome of a random (probabilistic) experiment which consists of the random selection of a unit from a population of interest (the underlying population of the random variable) and the recording of a numerical description of a characteristic(s) of the selected unit. In this chapter we will give a slightly more precise and detailed definition of a random variable. Moreover, we will give concise descriptions of the proba bility distribution of a random variable, which can also be thought of as the probability distribution on its sample space population, and will introduce several parameters of the probability distribution that are often used for descriptive and inferential purposes. The above concepts will be introduced separately for the two main categories of random vari ables, discrete random variables and continuous random variables. This chapter focuses on univariate variables. Similar concepts for bivariate and multivariate random variables will be discussed in the next chapter. 2 Random Variables In Section 2.2, it was pointed out that different variables can be associated with the same experiment, depending on the objective of the study and/or preference of the investigator; see Example B of that section for an illustration. The following examples reinforce this point and motivate a more precise definition of random variable . Example 2.1. Consider the experiment where two fuses are examined for the presence of a defect. Depending on the objective of the investigation the variable of interest can be the number of defective fuses among the two that are examined. Call this random variable X . Thus, the sample space corresponding to X is S X = { , 1 , 2 } . Alternatively, the investigation might focus on the whether or not the number of defective items among the two that are examined is zero. Call Y the random variable that takes the value zero when there are zero defective fuses, and the value 1 when there is at least one defective 1 fuse, among the two that are examined. Thus, the sample space that corresponds to Y is S Y = { , 1 } . Note that both X and Y are functions of the outcome in the basic, most detailed, sample space of the experiment, namely S = {DD , ND , DN , NN} . Example 2.2. In accelerated life testing, products are operated under harsher conditions than those encountered in real life. Consider the experiment where one such product is tested until failure, and let X denote the time to failure. The sample space of this experiment, or of X , is S X = [0 , ). However, if the study is focused on whether or not the product lasts more than, say, 1500 hours of operation, the investigator might opt to record a coarser random variable, Y , which takes the value 1 if the product lasts more than 1500 hours and the value 0 if it does not. The sample space of Y is S Y = {...
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This note was uploaded on 01/11/2012 for the course STAT 401 taught by Professor Akritas during the Fall '00 term at Pennsylvania State University, University Park.
 Fall '00
 Akritas

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