Technology & Engineering
Continuous and Discrete Random Variables
Continuous and discrete random variables are fundamental concepts in probability and statistics. Continuous random variables can take any value within a certain range, while discrete random variables can only take specific, distinct values. In technology and engineering, understanding the characteristics and behaviors of these random variables is crucial for modeling and analyzing various systems and processes.
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9 Key excerpts on "Continuous and Discrete Random Variables"
No longer available |Learn more- Jessica Utts, Robert Heckard(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
D E F I N I T I O N A random variable assigns a number to each outcome of a random circumstance. Equivalently, a random variable assigns a number to each unit in a population. Copyright 2014 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. 266 Chapter 8 Classes of Random Variables The two different broad classes of random variables follow: 1. Discrete random variables 2. Continuous random variables An example of a discrete random variable is the number of people with type O blood in a sample of ten individuals. The possible values are 0, 1, ... , 10, a list of distinct values. An example of a continuous random variable is height for adult women. With accurate measurement to any number of decimal places, any height is possible within the range of possibilities for heights. Between any two heights, there always are other possible heights, so possible heights fall on an infinite continuum. EXAMPLE 8.1 Random Variables at an Outdoor Graduation or Wedding Suppose you are par-ticipating in a major outdoor event, such as a graduation or wedding ceremony. Several random factors will determine how enjoyable the event will be, such as the temperature and the number of airplanes that fly overhead during the important speeches. In this context, temperature is a continuous random variable because it can take on any value in an interval. We often round off continuous random vari-ables to the nearest whole number, as with temperature in degrees, but conceptu-ally the value can be anything in an interval.
No longer available |Learn more- Anthony Hayter(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 2011 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. 74 CHAPTER 2 RANDOM VARIABLES FIGURE 2.4 X = positive difference between the scores of two dice 3 4 2 0 1 5 (5, 5) (6, 6) (5, 6) (6, 5) (4, 5) (4, 6) (4, 4) (6, 4) (5, 4) (3, 4) (3, 6) (3, 5) (3, 3) (6, 3) (5, 3) (4, 3) (2, 3) (6, 2) (5, 2) (4, 2) (3, 2) (2, 6) (2, 5) (2, 4) (2, 2) (1, 2) (1, 6) (1, 5) (1, 4) (1, 3) (1, 1) (6, 1) (5, 1) (4, 1) (3, 1) (2, 1) 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 X S The examples given above all concern discrete random variables as opposed to continuous random variables, which are discussed in the next section. A continuous random variable is one that may take any value within a continuous interval. For example, a random variable that can take any value between 0 and 1 is a continuous random variable. Mathematically speaking, continuous random variables can take uncountably many values. In contrast, discrete random variables can take only certain discrete values, as their name suggests. There may be only a finite number of values, as in Examples 1, 4, and 12, or infinitely many values, as in Example 13. The distinction between discrete and continuous random variables can most easily be understood by comparing the previous examples with the examples of continuous random variables in the next section.
eBook - PDFSix Sigma and Beyond
Statistics and Probability, Volume III
- D.H. Stamatis(Author)
- 2002(Publication Date)
- CRC Press(Publisher)
235 Discrete and Continuous Random Variables A special terminology is useful for discussing probability distributions of numerical scores. In this chapter we will discuss discrete and continuous random variables with some associated distributions. Some of the material in this chapter is used with permission from Stuart, A., Ford Motor Company. INTRODUCTION A random variable (RV) is a real valued number assigned to each individual sample, S i , according to some function X(S i ) Individual sample, S i Discrete real-valued number, X i . An individual sample cannot be assigned two random values. Range of real-numbers is typically finite. x = {X i } i = 1, 2, 3, …, M or x = {X 1, X 2, X 3 , …, X M } Assume an ascending order of magnitude: X i–1 < X i < X i+1 . Pictorially, this may be shown as: 16 Sample Space Function Processor X(S i ) = X i 0 Random Variables X i S i 236 Six Sigma and Beyond: Statistics and Probability, Volume III SAMPLES ASSIGNED THE SAME RANDOM VARIABLE 1. Two or more individual samples may be assigned the same numerical random value X i . 2. The total number of random variables M will be less than or equal to the total number of individual samples N. 3. Range of random variables: [X 1 £ X i £ X M ] (There are M discrete random variables in the range.) 4. If all samples are assigned a unique RV, then M = N. This may be shown in a pictorial form as: RANDOM VARIABLES GROUPED INTO CELLS 1. Group random variables into cells (or “bins”) if the number of random variables is large, say M > 30. 2. Groups or cells are formed in the numerical RV domain by dividing the range into say 10 to 20 equal cells. 3. While the total number of RVs is M, the number of cells is K. 4. All the individual RVs in a cell assume the value of the RV in, say the center of the cell interval and are denoted X k . 5. Each cell or grouping is designated X k and represents a finite interval of contiguous RVs.
- Charles Therrien, Murali Tummala(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
3 Random Variables and Transformations Situations involving probability do not always deal strictly with events. Frequently there are real-valued measurements or observations associated with a random experi-ment. Such measurements or observations are represented by random variables . This chapter develops the necessary mathematical tools for the analysis of experi-ments involving random variables. It begins with discrete random variables, i.e., those random variables that take on only a discrete (but possibly countably infinite) set of possible values. Some common types of discrete random variables are described that are useful in practical applications. Moving from the discrete to the continuous, the chapter discusses random variables that can take on an uncountably infinite set of possible values and some common types of these random variables. The chapter also develops methods to deal with problems where one random variable is described in terms of another. This is the subject of “transformations.” The chapter concludes with two important practical applications. The first involves the detection of a random signal in noise. This problem, which is fundamental to every radar, sonar, and communication system, can be developed using just the information in this chapter. The second application involves the classification of objects from a number of color measurements. Although the problem may seem unrelated to the detection problem, some of the underlying principles are identical. 3.1 Discrete Random Variables Formally, a random variable is defined as a function X ( · ) that assigns a real number to each elementary event in the sample space. In other words, it is a mapping from the sample space to the real line (see Fig 3.1). A random variable therefore takes on S s Mapping, X ( s ) = x real number line x Figure 3.1 Illustration of a random variable. a given numerical value with some specified probability. A simple example is useful to make this abstract idea clearer.
eBook - PDF- L. Z. Rumshiskii(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
R A N D O M V A R I A B L E S A N D P R O B A B I L I T Y D I S T R I B U T I O N S 31 § 8. C O N T I N U O U S R A N D O M V A R I A B L E S Discrete random variables do not form the only type of ran-dom variable: in probability theory we often encounter random variables whose ranges of possible values are entire intervals. For example, in the Introduction we discussed the problem of meas-uring the deviation from a central value of the dimensions of machined components. Such random variables are called continu-ous.^ The events of interest to us are of the form (x ± < ξ < x 2 ), and the probability distribution of ξ must enable us to find the proba-bility of this event for any interval (x 1 , x 2 ) . We shall denote this probability by F {x t < ξ < x 2 ). EXAMPLE . The uniform distribution. In the simplest case the range of a random variable is a finite interval ( α ι , α 2 ) , and if (*i 9 Xz) is any interval lying inside {OL 1 , a 2 ) , then the probability J*{xi < ξ < x 2 } is proportional to the length of this interval: Ρ{χ ί < ξ < x 2 } = λ(χ 2 - χ,) (<χ ί ύΧι<Χ 2 ύ « 2 ) . (2-16) We must choose the coefficient λ so that the second fundamental property of probabilities is satisfied;* since the range of all possible values of ξ is the interval ( a l s a 2 ) , the event (
- V. S. Pugachev(Author)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER 2 RANDOM VARIABLES Publisher Summary This chapter describes random variables. A random variable is a variable that assumes, as a result of a trial, only one of the set of possible values and with which is connected some field of events representing its occurrences in given sets, contained in the main field of events δ. Random variables may be both scalar and vector. In correspondence with general definition of a vector, one can call a vector random variable or a random vector any ordered set of scalar random variables. A random variable with countable or finite set of possible values is called a discrete random variable. The distribution of a discrete random variable is completely determined by the probabilities of all of its possible values. It is impossible to determine the distribution of a random variable with an uncountable set of possible values by the probabilities of its values. Therefore, another approach to such random variables is necessary. 2.1 General definitions. Discrete random variables 2.1.1 Definition of a random variable In Section 1.2.1 an intuitive definition of a random variable was given based on experimentally observable facts, and it was shown that with every random variable may be connected some events, its occurrences in different sets. For studying random variables it is necessary that the probabilities be determined for some set of such events, i.e. that this set of events belongs to the field of events δ connected with a trial. Furthermore, it is expedient to require that this set of events be itself a field of events (a subfield of the field δ)
eBook - PDF- Jingmei Jiang(Author)
- 2022(Publication Date)
- Wiley(Publisher)
91 Applied Medical Statistics , First Edition. Jingmei Jiang. © 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc. Companion website: www.wiley.comgojiangappliedmedicalstatistics We now consider another major class of random variables: the continuous variable. Unlike a count that is common in the discrete random variable, the continuous random variable can assume any value over an entire interval. Recall the example of the height of 10-year-old girls in Table 2.1, where the measurement value of height was kept to one decimal place so that the measured values were discrete. Assume that we improve the measurement precision to a theoretical maximum. The value of height can be any real number in a certain value range. Similarly, assume there is no measurement error, the measured values of physiological indices, for example, body weight, vital capacity, and serum test indices can be regarded as the collection of infinite points in a value range. These variables, in contrast to discrete random vari-ables, are called continuous random variables. However, because all possible values of a continuous random variable fill an interval a b , , it is impossible to assign finite probabilities to uncountable points in the interval in this manner. Therefore, the probability distribution of a continuous random variable can no longer be presented in the form of a probability mass function. Instead, a probability density function should be used. We start to discuss this new concept using the example of girls’ height.
eBook - PDFThe Probability Lifesaver
All the Tools You Need to Understand Chance
- Steven J. Miller(Author)
- 2017(Publication Date)
- Princeton University Press(Publisher)
This is why continuous random variables are often easier to handle, and more desirable as a closed form expression allows us to see how the answer changes as we vary the parameters. This chapter is preparatory for our discussion of many of the common continuous random variables in later chapters. We’ll describe some of their uses and many of their properties. To make those chapters self-contained, we’ll often go through identical arguments as we did in the discrete random variables chapters. It’s not bad seeing these arguments multiple times. The general framework is the same; the only difference is that we’ll have integrals to evaluate and not sums. Before getting to these continuous probability distributions however, we’ll first quickly review some results from calculus and other needed material in this chapter. Without fail, in almost every math class the part that gives students the greatest heartache is the material assumed known from earlier classes. In probability this is especially dangerous, as sometimes it has been a year (or years!) since a student has seen derivatives and integrals. Fortunately a brief refresher course is usually enough. If you would like a more detailed review of these concepts, I urge you to read Adrian Banner’s The Calculus Lifesaver , where all this material is carefully worked out. To gauge how well you remember your techniques of differentiation and integration, I’ve written Introduction to Continuous Random Variables • 239 and solved over 50 calculus problems; these (first the statements and then detailed solutions) and other supplementary material (such as a review of the Change of Variables Theorem and some calculus review lectures) are available in the online supplements at http://web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/. The plan of attack for this chapter is to 1. review the Fundamental Theorem of Calculus and its applications to probability, and then 2.
eBook - PDF- Roxy Peck, Chris Olsen, , Tom Short, Roxy Peck, Chris Olsen, Tom Short(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
For example, if the first magazine arrives on Friday and the second magazine arrives on Wednesday, then y 2. If both magazines arrive on Thursday, y 1. Determine the probability distribution of y . (Hint: Draw a tree diagram with two generations of branches, the first labeled with arrival days for Magazine 1 and the second for Magazine 2.) A continuous random variable is one that has possible values that form an entire interval on the number line. One example of a continuous random variable is the weight x (in pounds) of a full-term newborn baby. Suppose for the moment that weight is recorded only to the nearest pound. Then a reported weight of 7 pounds would be used for any weight greater than or equal to 6.5 pounds and less than 7.5 pounds. The probability distribution of x can be pictured as a probability histogram with rectangles centered at 4, 5, and so on. The area of each rectangle represents the probability of the corresponding weight value, and the total area of all the rectangles is 1. The probability that a weight (to the nearest pound) is between two values, such as 6 and 8, is the sum of the areas of the corresponding rectangles. Figure 7.5(a) illustrates this. Now suppose that weight is measured to the nearest tenth of a pound. There are many more possible reported weight values than before, such as 5.0, 5.1, 5.7, 7.3, and 8.9. As shown in Figure 7.5(b), the rectangles in the probability histogram are much narrower, and this histogram has a much smoother appearance. Now think about what would happen if weight were measured with greater accuracy. The probability histogram will look more like the smooth curve shown in Figure 7.5(c). This curve does not go below the horizontal measurement scale, and the total area under the curve is 1 (because this is true for all probability histograms). The probability that x falls in an interval such as 6 # x # 8 is represented by the area under the curve and above that interval.
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