Mathematics
Discrete Random Variable
A discrete random variable is a variable that can take on a countable number of distinct values. These values are typically the result of counting or enumerating, such as the number of students in a class or the outcomes of rolling a die. The probability distribution of a discrete random variable can be described using a probability mass function.
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eBook - PDF- Vladimir I. Rotar(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 3 Discrete Random Variables 1,2 1 RANDOM VARIABLES AND VECTORS 1.1 Random variables and their distributions Loosely put, a random variable (r.v.) is a variable whose values are determined “by chance.” More precisely, the value of a r.v. depends on which elementary outcome has occurred when the experiment under consideration has been performed. For example, when tossing a coin twice, we may be interested merely in the number of heads, no matter in which order they appeared. Possible results are given in the table below. Outcomes The number of heads ω 1 = { HH } 2 ω 2 = { HT } 1 ω 3 = { TH } 1 ω 4 = { TT } 0 So, while there are four outcomes, the number of heads may take on three values: either 0, or 1, or 2. We also see that the number of heads is completely determined by the outcome occurred. In the general case, when considering an experiment with a sample space Ω = { ω } , we de fi ne a random variable as a function X ( ω ) on the space Ω . EXAMPLE 1. In the case of rolling two dice, the sample space Ω = { ω } consists of 36 outcomes. The r.v. X ( ω ) , the sum of the numbers on the dice, may assume all integer values from 2 to 12. EXAMPLE 2. We choose at random a point from the disk { ( x , y ) : x 2 + y 2 ≤ 1 } ; see Example 1 .1.1-7. An elementary outcome ω may be identified with a point ( x , y ) from the disk. The distance from the point chosen to the center of the disk is the r.v. X ( ω ) = x 2 + y 2 which may take on any value from the interval [ 0 , 1 ] . As a rule, to simplify the notation, we will write X for X ( ω ) . A r.v. is said to be discrete if it can take on only a finite or countable number of possible values. If it is not the case, we call the r.v. non-discrete . Another term in use is a continuous r.v. , but it is slightly misleading; see Example 6 below. EXAMPLE 3. The number of successes in n trials is a discrete r.v.
eBook - PDF- Prem S. Mann(Author)
- 2020(Publication Date)
- Wiley(Publisher)
Thus, this value depends on the outcome of a random experiment. Consequently, x is called a random variable or a chance variable. In general, a random variable is denoted by x or y. Random Variable A random variable is a variable whose value is determined by the outcome of a random experiment. As will be explained next, a random variable can be discrete or continuous. 5.1 Random Variables 199 5.1.1 Discrete Random Variable A Discrete Random Variable assumes values that can be counted. In other words, the consec- utive values of a Discrete Random Variable are separated by a certain gap. Discrete Random Variable A random variable that assumes countable values is called a Discrete Random Variable. In Table 5.1, the number of vehicles owned by a family is an example of a Discrete Random Variable because the values of the random variable x are countable: 0, 1, 2, 3, and 4. Here are a few other examples of Discrete Random Variables: 1. The number of cars sold at a dealership during a given month 2. The number of houses in a certain block 3. The number of fish caught on a fishing trip 4. The number of complaints received at the office of an airline on a given day 5. The number of customers who visit a bank during any given hour 6. The number of heads obtained in three tosses of a coin 5.1.2 Continuous Random Variable A random variable whose values are not countable is called a continuous random variable. A continuous random variable can assume any value over an interval or intervals. Continuous Random Variable A random variable that can assume any value contained in one or more intervals is called a continuous random variable. Because the number of values contained in any interval is infinite, the possible number of values that a continuous random variable can assume is also infinite. Moreover, we cannot count these values. Consider the life of a battery.
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
Thus, this value depends on the outcome of a random experiment. Consequently, x is called a random variable or a chance variable. In general, a random variable is denoted by x or y. Random Variable A random variable is a variable whose value is determined by the outcome of a random experiment. As will be explained next, a random variable can be discrete or continuous. 5.1.1 Discrete Random Variable A Discrete Random Variable assumes values that can be counted. In other words, the consecutive values of a Discrete Random Variable are separated by a certain gap. Discrete Random Variable A random variable that assumes countable values is called a Discrete Random Variable. In Table 5.1, the number of vehicles owned by a family is an example of a Discrete Random Variable because the values of the random variable x are countable: 0, 1, 2, 3, and 4. Here are a few other examples of Discrete Random Variables: 1. The number of cars sold at a dealership during a given month 2. The number of houses in a certain block 5.1 Random Variables 5.1 Random Variables 181 3. The number of fish caught on a fishing trip 4. The number of complaints received at the office of an airline on a given day 5. The number of customers who visit a bank during any given hour 6. The number of heads obtained in three tosses of a coin 5.1.2 Continuous Random Variable A random variable whose values are not countable is called a continuous random variable. A continuous random variable can assume any value over an interval or intervals. Continuous Random Variable A random variable that can assume any value contained in one or more intervals is called a continuous random variable. Because the number of values contained in any interval is infinite, the possible number of values that a continuous random variable can assume is also infinite. Moreover, we cannot count these values. Consider the life of a battery.
eBook - PDFProbability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers
- Roy D. Yates, David J. Goodman(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
3 Discrete Random Variables 3.1 Definitions A random variable assigns numbers to outcomes in the sample space of an experiment. Chapter 1 defines a probability model. It begins with a physical model of an experiment. An experiment consists of a procedure and observations. The set of all possible observations, S, is the sample space of the experiment. S is the beginning of the mathematical probability model. In addition to S, the mathematical model includes a rule for assigning numbers between 0 and 1 to sets A in S. Thus for every A ⊂ S, the model gives us a probability P[A], where 0 ≤ P[A] ≤ 1. In this chapter and for most of the remainder of this book, we examine probability models that assign numbers to the outcomes in the sample space. When we observe one of these numbers, we refer to the observation as a random variable . In our notation, the name of a random variable is always a capital letter, for example, X. The set of possible values of X is the range of X. Since we often consider more than one random variable at a time, we denote the range of a random variable by the letter S with a subscript that is the name of the random variable. Thus S X is the range of random variable X, S Y is the range of random variable Y , and so forth. We use S X to denote the range of X because the set of all possible values of X is analogous to S, the set of all possible outcomes of an experiment. A probability model always begins with an experiment. Each random variable is related directly to this experiment. There are three types of relationships. 1. The random variable is the observation. Example 3.1 The experiment is to attach a photo detector to an optical fiber and count the number of photons arriving in a one-microsecond time interval. Each observation 53 54 CHAPTER 3 Discrete Random VariableS is a random variable X. The range of X is S X = {0, 1, 2, . . .}. In this case, S X , the range of X, and the sample space S are identical.
eBook - PDF- Eugene Demidenko(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Chapter 1 Discrete Random Variables Two types of random variables are distinguished: discrete and continuous. Theo- retically, there may be a combination of these two types, but it is rare in practice. This chapter covers discrete distributions and the next chapter will cover contin- uous distributions. 1.1 Motivating example In univariate calculus, a variable takes values on the real line and we write ∈ (−∞ ∞) In probability and statistics, we also deal with variables that take values in (−∞ ∞) Unlike calculus, we do not know exactly what value it takes. Some values are more likely and some values are less likely. These variables are called random. The idea that there is uncertainty in what value the variable takes was uncomfortable for mathematicians at the dawn of the theory of probability, and many refused to recognize this theory as a mathematical discipline. To convey information about a random variable, we must specify its distribution and attach a probability or density for each value it takes. This is why the concept of the distribution and the density functions plays a central role in probability theory and statistics. Once the density is specified, calculus turns into the principal tool for treatment. Throughout the book we use letters in uppercase and lowercase with different meaning: denotes the random variable and denotes a value it may take. Thus = indicates the event that random variable takes value For example, we may ask what is the chance (probability) that takes value In mathematical terms, Pr( = ) For a continuous random variable, we may be interested in the probability that a random variable takes values less or equal to or takes values from the interval [ + ∆] A complete coverage of probability theory is beyond the scope of this book — Advanced Statistics with Applications in R, First Edition. Eugene Demidenko. c ° 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc. 1
eBook - PDFProbability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers
- Roy D. Yates, David J. Goodman(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Chapter 4 covers continuous random variables. Definition 3.2 Discrete Random Variable X is a Discrete Random Variable if the range of X is a countable set S X = {x 1 , x 2 , . . .} . The defining characteristic of a Discrete Random Variable is that the set of possible values can (in principle) be listed, even though the list may be infinitely long. Often, but not always, a Discrete Random Variable takes on integer values. An exception is the random variable related to your probability grade. The experiment is to take this course and observe your grade. At Rutgers, the sample space is S = F, D, C, C + , B, B + , A . (3.5) We use a function G 1 (·) to map this sample space into a random variable. For example, G 1 (A) = 4 and G 1 (F ) = 0. The table Outcomes F D C C + B B + A G 1 0 1 2 2.5 3 3.5 4 is a concise description of the entire mapping. G 1 is a Discrete Random Variable with range S G1 = {0, 1, 2, 2.5, 3, 3.5, 4}. Have you thought about why we transform letter grades to numerical values? We believe the principal reason is that it allows us to compute averages. This is also an important motivation for creating random variables by assigning numbers to the outcomes in a sample space. Unlike probability models defined on arbitrary sample spaces, random variables have expected values , which are closely related to averages of data sets. We introduce expected values formally in Section 3.5. Quiz 3.1 A student takes two courses. In each course, the student will earn either a B or a C . To calculate a grade point average (GPA), a B is worth 3 points and a C is worth 2 points. The student’s GPA G 2 is the sum of the points earned for each course divided by 2. Make a table of the sample space of the experiment and the corresponding values of the GPA, G 2 . 3.2 Probability Mass Function The PMF of random variable X expresses the probability model of an experiment as a mathematical function. The function is the probability P[X = x] for every number x.
No longer available |Learn more- Anthony Hayter(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
C H A P T E R T W O Random Variables After the general discussion of probability theory presented in Chapter 1, random variables are introduced in this chapter. They are one of the fundamental building blocks of probability theory and statistical inference. The basic probability theory developed in Chapter 1 is extended for outcomes of a numerical nature, and distinctions are drawn between Discrete Random Variables and continuous random variables. Useful properties of random variables are discussed, and summary measures such as the mean and variance are considered, together with combinations of several random variables. Specific examples of common families of random variables are given in Chapters 3, 4, and 5. 2.1 Discrete Random Variables 2.1.1 Definition of a Random Variable A random variable is formed by assigning a numerical value to each outcome in the sample space of a particular experiment. The state space of the random variable consists of these numerical values. Technically, a random variable can be thought of as being generated from a function that maps each outcome in a particular sample space onto the real number line R , as illustrated in Figure 2.1. Random Variables A random variable is obtained by assigning a numerical value to each outcome of a particular experiment. A random variable is therefore a special kind of experiment in which the outcomes are numerical values, either positive or negative or possibly zero. Sometimes the experimental outcomes are already numbers, and then these may just be used to define the random variable. In other cases the experimental outcomes are not numerical, and then a random variable may be defined by assigning “scores” or “costs” to the outcomes. Example 1 Machine Breakdowns The sample space for the machine breakdown problem is S = { electrical, mechanical, misuse } and each of these failures may be associated with a repair cost .
eBook - PDF- Prem S. Mann(Author)
- 2017(Publication Date)
- Wiley(Publisher)
The number of houses in a certain block 3. The number of fish caught on a fishing trip A random variable is a variable whose value is determined by the outcome of a random experiment. A random variable that assumes countable values is called a Discrete Random Variable. 5.1 Random Variables 174 Chapter 5 Discrete Random Variables and Their Probability Distributions 4. The number of complaints received at the office of an airline on a given day 5. The number of customers who visit a bank during any given hour 6. The number of heads obtained in three tosses of a coin 5.1.2 Continuous Random Variable A random variable whose values are not countable is called a continuous random variable. A continuous random variable can assume any value over an interval or intervals. Because the number of values contained in any interval is infinite, the possible number of values that a continuous random variable can assume is also infinite. Moreover, we cannot count these values. Consider the life of a battery. We can measure it as precisely as we want. For instance, the life of this battery may be 40 hours, or 40.25 hours, or 40.247 hours. Assume that the maximum life of a battery is 200 hours. Let x denote the life of a randomly selected battery of this kind. Then, x can assume any value in the interval 0 to 200. Consequently, x is a continu- ous random variable. As shown in the diagram below, every point on the line representing the interval 0 to 200 gives a possible value of x. 0 200 Every point on this line represents a possible value of x that denotes the life of a battery. There is an infinite number of points on this line. The values represented by points on this line are uncountable. The following are a few other examples of continuous random variables: 1. The length of a room 2. The time taken to commute from home to work 3.
- John A. Gubner(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
2 Introduction to Discrete Random Variables In most scientific and technological applications, measurements and observations are expressed as numerical quantities. Traditionally, numerical measurements or observations that have uncertain variability each time they are repeated are called random variables . We typically denote numerical-valued random quantities by uppercase letters such as X and Y . The advantage of working with numerical quantities is that we can perform mathematical operations on them such as X + Y , XY , max ( X , Y ) , and min ( X , Y ) . For example, in a telephone channel the signal X is corrupted by additive noise Y . In a wireless channel, the signal X is corrupted by fading (multiplicative noise). If X and Y are the traffic rates at two different routers of an Internet service provider, it is desirable to have these rates less than the router capacity, say c ; i.e., we want max ( X , Y ) ≤ c . If X and Y are sensor voltages, we may want to trigger an alarm if at least one of the sensor voltages falls below a threshold v ; e.g., if min ( X , Y ) ≤ v . See Figure 2.1. + X+Y X Y + XY X Y X Y alarm status X Y X,Y max( ) c < _ output status OK if v < _ Trigger alarm if min( ) X,Y Figure 2.1. Systems represented by operations on random variables. In order to exploit the axioms and properties of probability that we studied in Chap-ter 1, we technically define random variables as functions on an underlying sample space Ω . Fortunately, once some basic results are derived, we can think of random variables in the traditional manner, and not worry about, or even mention the underlying sample space. This chapter introduces the student to random variables. Emphasis is on Discrete Random Variables and basic concepts and tools for working with them, such as probability mass functions, expectation, and moments. 2.1 Probabilities involving random variables A real-valued function X ( ω ) defined for points ω in a sample space Ω is called a ran-dom variable .
- Michael W. Trosset(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Chapter 4 Discrete Random Variables Our introduction of random variables in Section 3.5 was completely general; i.e., the principles that we discussed apply to all random variables. In this chapter we will study an important special class of random variables, the Discrete Random Variables. One of the advantages of restricting attention to Discrete Random Variables is that the mathematics required to define various fundamental concepts for this class is fairly minimal. 4.1 Basic Concepts We begin with a formal definition. Definition 4.1 A random variable X is discrete if and only if X ( S ) , the set of possible values of X , is countable. Our primary interest will be in random variables for which X ( S ) is finite; however, there are many important random variables for which X ( S ) is denumerable. The methods described in this chapter apply to both possi-bilities. In contrast to the cumulative distribution function (cdf) defined in Sec-tion 3.5, we now introduce the probability mass function (pmf). Definition 4.2 Let X be a Discrete Random Variable. The probability mass function (pmf ) of X is the function f : → defined by f ( x ) = P ( X = x ) . If f is the pmf of X , then f necessarily possesses several properties worth noting: 89 90 CHAPTER 4. Discrete Random VariableS 1. f ( x ) ≥ 0 for every x ∈ . 2. If x ∈ X ( S ), then f ( x ) = 0. 3. By the definition of X ( S ), x ∈ X ( S ) f ( x ) = x ∈ X ( S ) P ( X = x ) = P ⎛ ⎝ x ∈ X ( S ) { x } ⎞ ⎠ = P ( X ∈ X ( S )) = 1 . There is an important relation between the pmf and the cdf. For each y ∈ , let L ( y ) = { x ∈ X ( S ) : x ≤ y } denote the values of X that are less than or equal to y . Then F ( y ) = P ( X ≤ y ) = P ( X ∈ L ( y )) = x ∈ L ( y ) P ( X = x ) = x ∈ L ( y ) f ( x ) . (4.1) Thus, the value of the cdf at y can be obtained by summing the values of the pmf at all values x ≤ y .
- Carl-Louis Sandblom(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
68 5. Discrete Random Variables have a probability of zero. We shall defer a detailed discussion of continuous random variables to the next chapter and now devote our attention to Discrete Random Variables. We saw in the previous chapter that the possible outcomes for the random experiment toss a coin three times were: (H,H,H), (H,H,T), (H,T,H), (H,T,T), (T,H,H), (T,H,T), (T,T,H), (T,T,T), each with a probability of 1/8. If we define the random variable X as the number of heads in three tosses, we know that the possible values of X are 0, 1, 2, 3. We can also find the proba-bilities for the events X = j, for j = 0, 1, 2, 3: P(X = 0) = P((T,T,T)) = 1/8 P(X = 1) = P((H,T,T), (T,H,T), (T,T,H)) = 3/8 P(X = 2) = P((H,H,T), (H,T,H), (T,H,H)) = 3/8 P(X = 3) = P((H,H,H)) = 1/8 For the random variable X, we have now calculated the values of its probability distribution function p x J) = P ( X = j ) . We shall also write p (j) when there can be no confusion which random variable we are referring to. We can graphically portray the probability distribution function of our random variable: p
eBook - PDF- Barbara Illowsky, Susan Dean(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. On average, how many schools would you expect to offer such courses? 290 Chapter 4 | Discrete Random Variables This OpenStax book is available for free at http://cnx.org/content/col11562/1.17 e. Find the probability that at most ten offer such courses. f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence. 96. Suppose that about 85% of graduating students attend their graduation. A group of 22 graduating students is randomly chosen. a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many are expected to attend their graduation? e. Find the probability that 17 or 18 attend. f. Based on numerical values, would you be surprised if all 22 attended graduation? Justify your answer numerically. 97. At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon. a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many are expected to not to use the foil as their main weapon? e. Find the probability that six do not use the foil as their main weapon. f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically. 98. Approximately 8% of students at a local high school participate in after-school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number who participated in after-school sports all four years of high school.
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