1.1.3: Making Scaled Copies
Lesson
Let's draw scaled copies.
Exercise \(\PageIndex{1}\): More or Less?
For each problem, select the answer from the two choices.
-
The value of \(25\cdot (8.5)\) is:
- More than 205
- Less than 205
-
The value of \((9.93)\cdot (0.984)\) is:
- More than 10
- Less than 10
-
The value of \((0.24)\cdot (0.67)\) is:
- More than 0.2
- Less than 0.2
Exercise \(\PageIndex{2}\): Drawing Scaled Copies
- Draw a scaled copy of either Figure A or B using a scale factor of \(3\).
- Draw a scaled copy of either Figure C or D using a scale factor of \(\frac{1}{2}\).
Exercise \(\PageIndex{3}\): Which Operations? (Part 1)
Diego and Jada want to scale this polygon so the side that corresponds to 15 units in the original is 5 units in the scaled copy.
Diego and Jada each use a different operation to find the new side lengths. Here are their finished drawings.
- What operation do you think Diego used to calculate the lengths for his drawing?
- What operation do you think Jada used to calculate the lengths for her drawing?
- Did each method produce a scaled copy of the polygon? Explain your reasoning.
Exercise \(\PageIndex{4}\): Which Operations? (Part 2)
Andre wants to make a scaled copy of Jada's drawing so the side that corresponds to 4 units in Jada’s polygon is 8 units in his scaled copy.
- Andre says “I wonder if I should add 4 units to the lengths of all of the segments?” What would you say in response to Andre? Explain or show your reasoning.
- Create the scaled copy that Andre wants. If you get stuck, consider using the edge of an index card or paper to measure the lengths needed to draw the copy.
Are you ready for more?
The side lengths of Triangle B are all 5 more than the side lengths of Triangle A. Can Triangle B be a scaled copy of Triangle A? Explain your reasoning.
Summary
Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor.
For example, to make a scaled copy of triangle \(ABC\) where the base is 8 units, we would use a scale factor of 4. This means multiplying all the side lengths by 4, so in triangle \(DEF\), each side is 4 times as long as the corresponding side in triangle \(ABC\).
Glossary Entries
Definition: Corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
Definition: Scale Factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).
Definition: Scaled Copy
A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
Practice
Exercise \(\PageIndex{5}\)
Here are 3 polygons.
Draw a scaled copy of Polygon A using a scale factor of 2.
Draw a scaled copy of Polygon B using a scale factor of \(\frac{1}{2}\).
Draw a scaled copy of Polygon C using a scale factor of \(\frac{3}{2}\).
Exercise \(\PageIndex{6}\)
Quadrilateral A has side lengths 6, 9, 9, and 12. Quadrilateral B is a scaled copy of Quadrilateral A, with its shortest side of length 2. What is the perimeter of Quadrilateral B?
Exercise \(\PageIndex{7}\)
Here is a polygon on a grid.
Draw a scaled copy of this polygon that has a perimeter of 30 units. What is the scale factor? Explain how you know.
Exercise \(\PageIndex{8}\)
Priya and Tyler are discussing the figures shown below. Priya thinks that B, C, and D are scaled copies of A. Tyler says B and D are scaled copies of A. Do you agree with Priya, or do you agree with Tyler? Explain your reasoning.
(From Unit 1.1.1)


1.5: The Crime Control and Due Process Models
The criminal justice system can be quite complicated, especially in the attempt to punish offenders for wrongs committed. Society expects the system to be efficient and quick, but the protection of individual rights and justice is fairly delivered. Ultimately, the balance of these goals is ideal, but it can be challenging to control crime and quickly punish offenders, while also ensuring our constitutional rights are not infringed upon while delivering justice.
In the 1960s, legal scholar Herbert L. Packer created models to describe exceeding expectations of the criminal justice system. These two models can be competing ideologies in criminal justice, but we will discuss how these models can be merged or balanced to work together. The first tension between these models is often the values they place as most important in the criminal justice system. These two models are referred to as the crime control model and the due process model.
The crime control model focuses on having an efficient system, with the most important function being to suppress and control crime to ensure that society is safe and there is public order. Under this model, controlling crime is more important than individual freedom. This model is a more conservative perspective. In order to protect society and make sure individuals feel free from the threat of crime, the crime control model advocates for swift and severe punishment for offenders. Under this model, the justice process may resemble an assembly-line. Law enforcement suspects; they apprehend suspects; the courts determine guilt; and guilty people receive appropriate, and severe, punishments through the correctionalsystem.The crime control model may be more likely to take a plea bargain because trials may take too much time and slow down the process.
Imagine you are working out at the local gym, and a man starts shooting people. This man has no mask on,so he is easy to identify. People call 911,police promptly respond,and arrest the shooter within minutes. Under the crime control model, the police should not have to worry too much about how evidence gets collected and expanded. Investigative, arrest,and search powers would be considered necessary. A crime control model would see no need to waste time or money by ensuring due process rights.
Legal technicalities, such as warrantless searches of the suspect's home, would obstruct the police from effectively controlling crime. Effective use of time would be to immediately punish, especially since the gym had cameras and the man did not attempt to hide his identity. Any risk of violating individual liberties would be considered secondary to the need to protect and ensure the safety of the community in this model. Additionally, the criminal justice system is responsible for ensuring victim’s rights, especially helping provide justice for those murdered at the gym.
Using the same example of the gym murder, the due process model would want to see all the formalized legal practices afforded to this case in order to hold him accountable for the shooting. If this man did not receive fair and equitable treatment, then the fear is this can happen to other cases and offenders. Therefore, due process wants the system to move through all the stages to avoid mistakes and ensure the rights of all suspects and defendants. If the man in the gym pled not guilty due to the reason of insanity, then he can ask for a jury trial to determine whether he is legally insane. The courts would then try the case and may present evidence to a jury, ultimately deciding hisfate.The goal is not to be quick, but to be thorough. Because the Bill of Rights protects the defendant’s rights, the criminal justice system should concentrate on those rights over the victim’s rights, which are not listed. Additionally,limiting police power would be seen as positive to prevent oppressing individuals and stepping on rights. The rules, procedures, and guidelines embedded in the Constitution should be the framework of the criminal justice system and controlling crime would be secondary. Guilt would get established on the facts and if the government legally followed the correct procedures. If the police searched the gym shooter's home without a warrant and took evidence then that evidence should be inadmissible, even if that means they cannot win the case.
There are several pros and cons to both models; however, there are certain groups and individuals that side with one more often than the other. The notion that these models may fall along political lines is often based on previous court decisions, as well as campaign approaches in the U.S. The crime control model is used when promoting policies that allow the system to get tough, expand police powers, change sentencing practices such as creating “Three Strikes,”and more. The due process model may promote policies that require the system to focus on individual rights. These rights may include requiring police to inform people under arrest that they do not have to answer questions without an attorney (Miranda v. Arizona), providing all defendants with an attorney (Gideon v. Wainwright), or shutting down private prisons that often abuse the rights of inmates.
To state that crime control is purely conservative and due process is purely liberal would be too simplistic, but it is relevant to recognize that the policies are a reflection of our current political climate. If Americans are fearful of crime, as Gallup polls suggest they are, politicians may propose policies that focus on controlling crime. However, if polls suggest police may have too many powers and that can lead to abuse, then politicians may propose policies that limit their powers such as requiring warrants to obtain drugs.Again, this may reflect society as a whole, a sector of society, or the interests of a political party or specific politician.


19.2: Personal Construct Theory
-
- Last updated
- Save as PDF
Kelly’s personal construct theory departs from cognitive social learning in that he proposes it is not simply enough to know what a person is likely to do in a given situation, even when your predictions are correct. More importantly, we need to know what a person might have done (Kelly, 1966). Thus, unlike the cognitive social learning theorists who consider cognitive processes as an aspect of the environmental circumstances associated with behavior, Kelly focused on the cognitive constructs first and foremost.
Kelly presented his personal construct theory in a two volume set, which was published in 1955 (Kelly, 1955a,b). In the mid-1960s Kelly was preparing a new book that was to include talks and papers he had presented around the world, many of which were never published. Unfortunately, he did not survive to complete this task, but the project was later completed by Brendan Maher (Maher, 1969), thus furthering the information available on this unique theory.
Brief Biography of George Kelly
Not much is known about George Kelly as a person, particularly regarding his childhood.  Thus, it has been somewhat difficult to piece together a picture of how this fascinating individual became the man he was. He was born in 1905 on a farm near Perth, Kansas. His father had been educated for the Presbyterian ministry, but after getting married his parents moved to the farm in Kansas, where they had their only child. The Kelly family moved around from farm to farm, including a failed farm in Colorado on some of the last free land given to settlers in the West. Kelly’s education was erratic, and he learned what he could when the family would occasionally spend a few weeks in town. He attended four different high schools, and apparently never established any long-term relationships. He was, however, favorably influenced by the exciting stories told by his maternal grandfather, who had been the captain of a sailing ship in the North Atlantic (Fransella, 1995).
Despite his erratic education, Kelly attended college at Friends University and then Park College, where he received a Bachelor’s degree in physics and mathematics. Despite studying math and science, the collegiate debates he experienced had sparked a keen interest in social problems. So, he entered the University of Kansas to earn a Master’s degree in educational sociology, and in 1927 he completed his thesis on the distribution of leisure time activities of workers in Kansas City. He then moved to Minneapolis and supported himself by teaching one night a week in each of three different night schools. He began studying sociology and biometrics at the University of Minnesota, but when the university found out that he couldn’t pay his tuition he was told to leave. He was then hired to teach psychology and speech at Sheldon Junior College in Iowa, where he spent a year and a half. He then returned to the University of Minnesota for a semester of studying sociology, but then went back to Wichita, Kansas and worked as an aeronautical engineer for a few months. He then received an exchange scholar fellowship to study at the University of Edinburgh in Scotland, where he earned a Bachelor’s degree in education in 1930, having written his thesis on predicting teaching success. He then returned to the United States, attended the State University of Iowa, and in 1931 he received a Ph.D. in psychology, with a focus on reading and speech disabilities. Two days later he married Gladys Thompson (Fransella, 1995; Maher, 1969).
As uncharacteristic as it may seem, Kelly finally settled down. He spent the next ten years teaching at Fort Hays Kansas State College. His research and writings during this time focused on the practical aspects of providing clinical psychological services for schools. Much like Alfred Adler had in Austria, he developed traveling clinics to provide training around the state of Kansas, and his model had a dramatic influence on the future of rural school psychology. With the advent of World War II, however, Kelly entered the Navy as an aviation psychologist. At first he helped to train local civilian pilots, but then he transferred to the Bureau of Medicine and Surgery of the Navy in Washington to help select naval air cadets. After the war he spent a year at the University of Maryland, but in 1946 he was appointed Professor and director of Clinical Psychology at Ohio State University, where he spent the next 20 years (Fransella, 1995; Maher, 1969).
During his first few years at Ohio State he focused on making their clinical psychology training program into one of the best in the country. One of his colleagues in the department was Julian Rotter, also known for his role in shaping the training programs used in psychology today, and one of their students was Walter Mischel (who admired both Rotter and Kelly). Kelly then turned his attention to the theory that made him famous, the two volume work entitled The Psychology of Personal Constructs (Kelly, 1955a,b). Kelly’s theory gained immediate recognition as both unique and significant, and he was invited to teach and lecture around the United States and in Europe, the Soviet Union, South America, Asia, and the Caribbean. He was elected president of both the Clinical Division and the Consulting Division of the American Psychological Association, and he served as president of the American Board of Examiners in Professional Psychology. In 1965, Kelly left Ohio State for Brandeis University, where he was appointed to the Riklis Chair of Behavioral Science. He began working on a collection of his papers and lectures from the past decade, but he died unexpectedly in March, 1966. As mentioned above, that collection of works was completed by the man who succeeded Kelly as Riklis Professor of Behavioral Science, and was published under the title Clinical Psychology and Personality: The Selected Papers of George Kelly (Maher, 1969).
Placing Kelly in Context: A Cognitive Theory of Personality
Simply put, Kelly’s personal construct theory represents the culmination of the shift from animalistic behaviorism to humanistic cognition. In American psychology, behaviorism was a powerful force, and began with the very traditional approach of theorists such as John B. Watson and B. F. Skinner. Alongside the experimental behaviorists were the learning theorists, such as Clark Hull. As Dollard and Miller tried to find some common ground between psychodynamic theory and traditional learning approaches, they were inevitably led to consider the role of social factors in human learning. Bandura, Rotter, and Mischel built on the legacy of Dollard and Miller, but added to it the active role of cognition in the human species. Finally, Kelly moved to a purely cognitive description of how individuals become who they are.
However, we must address an important caveat. Just as Skinner’s radical behaviorism was an extreme position of ignoring cognitive processes, perhaps Kelly’s position is equally extreme for attributing significant cognitive processing to all aspects of personality and personality development. Although the man-the-scientist concept may hold a certain curious appeal for some, who doesn’t sometimes knowingly try acting in different ways to see what effect it has on others? Likewise, who isn’t attracted to a theory that says we desire the ability to both predict and control the events in our lives? However, as Kelly the therapist was keenly aware, many people cannot predict or control the events surrounding them. Is this always the result of failed construct systems, or is it possible that sometimes we just aren’t thinking? Regardless of the answer, Kelly is the recognized leader of a significant development in the field of personality, a development that contributed to the highly regarded cognitive therapies of Ellis and Beck.
It is also interesting to note that the basis for his theory of constructive alternativism questions the reality of the self in a manner similar to Eastern/Buddhist concepts of consciousness and self. Carl Jung was dramatically influenced by the ancient Vedic traditions of India, and Carl Rogers, the founder of humanistic psychology, was influenced by the spiritual traditions of China. And now we have Kelly, whose theory represents the culmination of behavioral-cognitive theories, sharing a fundamental similarity with Buddhist psychology. Clearly, throughout the history of personality theory, there have been important theorists who looked beyond the constraints of their own training and their own culture.
Constructive Alternativism
Kelly begins by questioning the role that psychologists have assigned to themselves. Psychologists consider themselves to be scientists, engaged in the systematic study of human behavior and thought. Kelly questions why we, as psychologists, don’t extend the same perspective to all people. According to Kelly, doesn’t every person seek to predict and control the course of events in their lives? Doesn’t every person have their own theories about situations in life, don’t they test their own hypotheses, and weigh the experimental evidence gained through experience?
…it is as though the psychologist were saying to himself, “I, being a psychologist , and therefore a scientist , am performing this experiment in order to improve the prediction and control of certain human phenomena; but my subject, being merely a human organism, is obviously propelled by inexorable drives welling up within him, or else he is in gluttonous pursuit of sustenance and shelter.” (pg. 5; Kelly, 1955a)
Since Kelly proposes that each individual is theorizing about and testing their own life circumstances, he suggests the term man-the-scientist for understanding how all people (including, of course, women) approach the world around them.
In trying to understand the world around us, using arguments that sound either existential or like Eastern philosophy, Kelly questions the existence of the universe. Of course it exists, he says, but so do the thoughts of each individual, and the correspondence between what people think exists and what does, in fact, exist is constantly changing. Thus, Kelly suggests that it is better to say that the world around us is existing , rather than to say it exists. Likewise, life can only be understood in the context of time, if it is to make sense. However, life is not simply the changes that occur over time. Rather, it is a relationship between living things and their environment. Kelly emphasizes the creative capacity of living things to represent their environment, as opposed to simply reacting to it. These representations are known as constructs , patterns that we create in our mind and attempt to fit over the realities of the world. Since our constructs don’t always fit with reality, we are constantly modifying them, as well as trying to increase our repertoire of constructs. Over time, we test our constructs for the ability to predict what will happen in our lives. With sufficient time and experience, and if we are willing to learn from our mistakes, we can evaluate all of our interpretations of the world in which we live (Kelly, 1955a).
Kelly believes that all of our present interpretations of the environment are open to revision or replacement; there are always alternative constructs that may help us deal with new or difficult situations. It is this philosophical position that Kelly refers to as constructive alternativism. It is important to keep in mind, of course, that not just any alternative will work in a given situation. Therefore, each potential alternative construct must be evaluated in terms of its specific predictive efficiency, as well as in terms of overall predictive efficiency of the system it would become part of, if that alternative construct were adopted (Kelly, 1955a).
discussion question \(\PageIndex{1}\)
According to Kelly, it is more important to know what a person might have done, and he believed that people act as scientists, testing their constructs in order to become better at predicting and controlling their lives. Can you think of situations in which someone did what you expected, but you really wanted to know what they had thought about doing as alternatives?
Basic Theory of Personal Constructs
Personal construct theory begins with a fundamental postulate , which is then elaborated with eleven corollaries. The fundamental postulate states that “ a person’s processes are psychologically channelized by the ways in which he anticipates events ” (Kelly, 1955a). The carefully chosen words in this postulate define the nature of personal construct theory. The words person and he emphasize the individuality of this theory, the unique nature of each person’s constructs. Each person is then recognized as a process. The mind does not stop and start, simply reacting to stimuli, but rather it is constantly in motion, constantly experimenting with constructs. These processes operate through a network of pathways, or channels , according to the devices, or ways , a person constructs in order to achieve their goals. Since these processes, ways, and channels have not been identified as specific physiological mechanisms or anatomical structures, Kelly emphasizes that this is a psychological theory. So when we discuss this mechanism we are not necessarily addressing neuroscience on one hand or sociology on the other hand, we are working within the constraints of the field of psychology. We then have anticipation , the “push and pull of the psychology of personal constructs” (Kelly, 1955a). Being man-the-scientist, each of us seeks to predict the future and choose our actions accordingly. Finally, we have real-life events . Kelly was always very practical about both his personality theory and his approach to psychotherapy. Thus, the psychology of personal constructs is not an ethereal theory. Psychological processes, according to Kelly, are tied down to reality, and anticipation is carried forward in order to better represent future reality. Having established the fundamental postulate, Kelly then described eleven corollaries, or propositions, which both follow from the postulate and amplify his system by elaborating on the fundamental idea (Kelly, 1955a).
Construction Corollary: A person anticipates events by construing their replications. Construing refers to placing an interpretation upon an event. Since a new event will not occur exactly as a past event, our anticipation involves interpreting what the new event will be like. Kelly uses the example of a day. Today is not the same as yesterday, tomorrow will not be the same as today, but each day follows something of a similar pattern. Thus, our anticipation of tomorrow involves constructs based on both the similarities and differences between days we have experienced in the past. It is important to note that this process is not the same as cognition, it is not simply thinking about tomorrow. Much of this process is preverbal, or unconscious, and in that sense occurs automatically.
Individuality Corollary: Persons differ from each other in their construction of events. No matter how closely associated two people are, they cannot play exactly the same role in any situation. Therefore, they will interpret events differently. Although Kelly acknowledges that people often share similar experiences, particularly as they attend to the experiences of others in the same or similar situations, this corollary emphasizes the unique, subjective nature of interpreting and anticipating events.
Organization Corollary: Each person characteristically evolves, for his convenience in anticipating events, a construction system embracing ordinal relationships between constructs. When faced with conflict, there may be solutions that contradict one another. Thus, the constructs we develop may contradict each other. Kelly suggested that we develop our constructs in a systematic and organized way, with some constructs being ranked more highly than others.   For instance, suppose you have an electric garage door opener, and the power is out. You could put your car in the garage by driving through the garage door. However, it might be preferable to get out of the car, go into the garage through the house or side door, and then disconnect the garage door from the opener and open it by hand.
Dichotomy Corollary: A person’s construction system is composed of a finite number of dichotomous constructs. Every construct has both positive and negative aspects. In the example used above, both ways of putting your car in the garage have advantages and disadvantage. Driving through the garage door is quick and easy in the short term, but results in needing a new garage door. Getting out of the car and using another entrance takes more time and effort, and may be unpleasant in a bad storm, but it protects your property (and saves time and money in the long run). The essential nature of contrast was eloquently described some 2,600 years ago by Lao Tsu:
Under Heaven all can see beauty as beauty only because there is ugliness.
All can know good as good only because there is evil.
Therefore having and not having arise together.
Difficult and easy complement each other.
Long and short contrast each other;
High and low rest upon each other;
Voice and sound harmonize each other;
Front and back follow one another.
Lao Tsu, c600 B.C.
(pg. 4; Lao Tsu, c600 B.C./1989)
Choice Corollary: A person chooses for himself that alternative in a dichotomized construct through which he anticipates the greater possibility for extension and definition of his system. Simply put, since each situation requires us to choose between the options we construct, Kelly believed we choose the alternative that serves us best (at least within our system of constructs, which may be different than the reality of the best choice). But what about situations in which the best choice is not so obvious? Kelly believed that the choice corollary allowed for shades of gray when a decision is not clearly a choice between black and white alternatives. He did not view this as a contradiction, but rather, he proposed that the choice becomes one between options that are more gray or less gray. Thus, we can maintain the dichotomy of the choice while still also allowing the choice itself.
Range Corollary: A construct is convenient for the anticipation of a finite range of events only. Every personal construct has a range or focus, and few, if any, are relevant to all events. As Kelly points out, the construct tall vs. short may apply well to descriptions of people, trees, and buildings. But what would we mean by tall weather, or short light? Clearly, the construct tall vs. short is limited to certain types of discrete, physical objects.
Experience Corollary: A person’s construction system varies as he successively construes the replications of events. As we apply constructs in our efforts to predict what happens in our lives, we sometimes experience unexpected outcomes. As a result, we reconstruct our constructs, and learn from our experiences. In other words, man-the-scientist is by definition a work in process, and that process is ongoing.
Modulation Corollary: The variation in a person’s construction system is limited by the permeability of the constructs within whose range of convenience the variants lie. This corollary addresses the ease with which the experience corollary can occur. Although all individuals modify the constructs that guide their anticipation of events, some constructs are modified more easily, and some people are more open to changing their perspectives (and, hence, reconstructing their constructs). This is one area where Kelly re-emphasizes the difference between psychological processes and the process of science. Scientists seek hypotheses, theories, and laws that are not likely to change. Indeed, there is a continuum from hypothesis to law based on how likely it is that a scientific observation is true. People, however, are constantly testing and retesting their constructs, and reconstructing them as necessary and appropriate. Therefore, people may act like scientists, but their psychological processes serve to facilitate the individual’s life, not the lives of others (as scientific theories and laws are meant to apply to the whole universe).
Fragmentation Corollary: A person may successively employ a variety of construction subsystems which are inferentially incompatible with each other. This corollary extends from the previous one, but with a twist. As an individual encounters unexpected events, they modify their constructs to the extent that they are able. Thus, their behavior may change slowly, or more quickly, depending on the nature of the constructs that guide their openness to change. The twist comes into play when individuals are either resisting change, or in the process of change, and it involves the dichotomy corollary. If an individual is failing to predict and control events in their life, they may choose an incompatible construct, essentially reversing the course of their behavior. One of the advantages of Kelly’s personal construct theory is that these dramatic changes in behavior can now be seen as reasonable progressions in one’s ongoing desire for predictability and control.
Commonality Corollary: To the extent that one person employs a construction of experience which is similar to that employed by another, his psychological processes are similar to those of the other person. This corollary is important for interpersonal relations. Even though two people cannot experience the same event in exactly the same way, their ability to share their experiences is facilitated by the similarity of their experiences. This raises important implications for therapists working with clients of different cultures, since they might not share similar constructs based on certain events. It also raises an important distinction between cognitive and behavioral approaches to understanding personality. In behavioral perspectives, simple stimulus-response relationships are the same for everyone who experiences them. However, in the cognitive perspective, each person necessarily experiences any event in a unique way.
Sociality Corollary: To the extent that one person construes the construction processes of another, he may play a role in a social process involving the other person. Humans are social creatures. Our ability to predict and control our lives is largely based on our ability to predict and either control or work with other people. Thus, it is important for individuals to construe to thoughts and behaviors of others, and in so doing we can each play a role in the lives of others. Kelly suggests that this provides a natural connection between personal construct theory and social psychology, as well as a connection between personal construct theory and cross-cultural psychology.
Dimensions of Transition
Since life is an ongoing process, there are regular transitions in one’s personal constructs. According to the organizational and modulation corollaries, individuals have certain preferences amongst their constructs and differences in their ability to reconstruct them. Problems in life arise when individuals find it difficult to transition from an ineffective construct, one that does not allow for predicting or controlling events, to an effective construct. According to Kelly (1955a), the major problems are seen as the psychological phenomena of threat, fear, guilt, and anxiety. Kelly defines these terms as follows:
Threat is the awareness of imminent comprehensive change in one’s core structures.
Fear is like threat, except that, in this case, it is a new incidental construct, rather than a comprehensive construct, that seems about to take over.
Perception of one’s apparent dislodgment from his core role structure constitutes the experience of guilt.
Anxiety is the recognition that the events with which one is confronted lie outside the range of convenience of one’s construct system.
In each case, the psychological phenomenon is based on either the failure of one’s constructs to provide courses of action or a direct challenge to the system of constructs available to the individual. Given that the individual’s personal construct systems define the larger construct of self, these psychological phenomena represent a challenge to the very self experiencing them. In defense of the individual, aggressiveness is seen as the active elaboration of one’s perceptual field . In other words, aggressive individuals try to control events in ways that force decisions favorable to the individual. Similarly, hostility is viewed as the continued effort to extort validational evidence in favor of a type of social prediction which has already proved itself a failure . In this case, the individual tries to find confirmation of success following failed constructs, and this can only be done at the expense of others (Kelly, 1955a).
As people live their everyday lives, there are two typical cycles of transition, the C-P-C Cycle and the Creativity Cycle . The C-P-C Cycle involves circumspection, preemption, and control. Being circumspect refers to being wary, not taking risks. Thus, as we construe events we try to be precise in the development of our constructs. We then preempt these constructs for membership in an exclusive realm, one that best fits the event we are trying to predict and control. Finally, the first two steps have control as their natural consequence. Still, the individual must make the choice of that course of action, so Kelly suggests that the final C could just as well stand for choice as it does for control. In contrast, the Creativity Cycle begins with loose constructions, and then leads to tightened and validated constructions. What makes the Creativity Cycle meaningful is the individual’s ability to quickly experiment with various constructs and then seize upon the most promising, which is then tightened up and tested. Since much of this process is preverbal, the thought processes of creative individuals may not be apparent to others. According to Kelly, although individuals who begin with tight constructions might be productive, they cannot be creative. Creativity requires beginning with loose constructions (Kelly, 1955a). The value of creativity is not simply to be found as a distinction between the types of cycles experienced by individuals in their daily lives. Creativity is an important component of well-being, and a common topic in books on positive psychology and human strengths and virtues (Aspinwall & Staudinger, 2003; Cloninger, 2004; Compton, 2005; Peterson & Seligman, 2004; Snyder & Lopez, 2005). Indeed, Carl Rogers identified creativity as a significant aspect of the personality of a fully functioning person (Rogers, 1961).
discussion question \(\PageIndex{2}\)
Many psychologists, including Kelly and Carl Rogers, consider creativity to be an essential aspect of healthy psychological development. Have you ever entered into an unfamiliar situation and tried to be creative in how you handled it? Was it difficult to set aside preconceived notions about how to act, or do you find it easy to try different things in new situations?
The Role of the Psychotherapist
Kelly was first and foremost a clinical psychologist, and his writings are full of practical examples related to clinical work. Theories are of little value unless they are rooted in the values of the psychologists using them. Kelly considered this to be particularly true of clinical psychologists, since they are routinely dealing with clients (Kelly, 1955b). Kelly believed that the role of the psychotherapist involves not only the training and perspectives of the therapist, but also a need to understand the client, and essential ethical considerations:
The role of the psychotherapist involves keen alertness to what the client expects from psychotherapy and the initial acceptance of a wide variety of client misperceptions of what psychotherapy is…it involves certain ethical obligations that transcend mere legal status. (pg. 618; Kelly, 1955b)
In his typically practical style, Kelly provides lengthy lists of what psychotherapy means to the client, the client’s conceptualization of the therapist, the clinician’s conceptualization of their role, and basic approaches to revising the client’s constructs. At the beginning of therapy, it is unlikely that the client has a good concept of what therapy is and what it can accomplish. Nonetheless, the client has some construction of what will take place in therapy. The complaint presented by the client says something about what he or she thinks therapy can accomplish, and some see therapy as an end in itself. However, the reality is that therapy is a means to an end, and that end should be generating movement forward on the part of the client. In contrast, some clients are so ready for change that the therapist must be cautious in interpreting the client’s state of mind (Kelly, 1955b). Just as the client makes predictions about therapy, they will also have an initial conceptualization of the therapist. They may construe the therapist as a parent, an absolver of guilt, a companion, or even a threat. Hopefully, according to Kelly, the client may construe the therapist as a representative of reality. In this case, the client may feel free to experiment with his or her constructs without fear of failure. Most importantly, how the client construes the therapist will have a dramatic effect on their relationship:
From the client’s conceptualization of psychotherapy comes the role he expects to play and the role he expects the therapist to play…He may be bitterly disappointed in the therapist’s enactment of the expected role. He may stretch his perceptions of the therapist in order to construe him in the manner he expected to construe him rather than in the manner the therapist seeks to be construed….The client may then feel lost and insecure in the psychotherapeutic relationship. (pg. 575; Kelly, 1955b)
As the client is engaging in these processes, the clinician is also conceptualizing their own role. Overall, the goal of any therapist should be to assist in the continuous reconstruction of the client’s construct system, and the changes that take place in therapy should set the stage for continued reconstruction after therapy has been discontinued. Initially, the therapist may rely on a variety of techniques to accomplish superficial reconstructions. The therapist needs to be patient, and initially must accept the client’s construct systems as they are. The latter point is quite similar to the empathy described by Carl Rogers in humanistic, client-centered therapy, and Kelly does indeed use the word “empathize” in his own writings. As therapy progresses, the therapist needs to help the client select new conceptual elements, accelerate the tempo of the client’s experience, and design and implement experiments. Finally, the therapist serves to validate the client’s experiments as they attempt to reconstruct their construct systems (Kelly, 1955b).
Psychological Assessment within Personal Construct Theory
Kelly believed that therapy was a joint effort between the therapist and the client, and since the goal was the ongoing reconstruction of the client’s psychological systems (even after therapy), the client ultimately needs to become his own therapist. Therefore, the psychotherapeutic interview (Note: by “interview,” Kelly means what we would commonly call a therapy session) becomes an essential part of therapy. Throughout the process of the interview, the therapist makes decisions regarding the course of the interchange between the therapist and the client. Overall, the decisions made by the therapist are tailored to the specific client, but still the therapist must remain in control of the interview. This requires that the therapist plan for the interview. Those plans include how often to interview the client, how long the interviews should last, the tempo of the interview, and when to terminate the interview. Since the client continues to live their life outside of the interview room, the therapist must also consider whether special circumstances will require special interview plans (Kelly, 1955b, 1958). One of the most practical aspects of the interview is that the client can simply provide information needed by the therapist, to a point:
…there is a useful adage for clinical psychologists to follow on occasion: if you do not know what is wrong with a person, ask him; he may tell you. (pp. 322; Kelly, 1955a)
The Role Construct Repertory Test (Rep Test) was developed by Kelly in order to understand how a client’s personal constructs influence their personal-social behavior. The client begins with a Role Title List , on which they list the names of important people in their lives (see Table 10-1). The names are then grouped three at a time, and the client is asked to describe in what important way two of the three individuals are alike but different than the third person. A more organized form of the Rep Test, particularly useful for research purposes, involves creating the Repertory Grid . Once again the client is asked to identify significant people in their life. The grid provides three-person pairings that address various relational factors (family, intimate friends, conflicted relationships, authority figures, and values), and as before the client provides a construct that associates two of the people yet distinguishes them from the third. The common factor is listed as the emergent pole, the distinguishing factor is listed as the implicit pole . The Rep Test does not result in specific outcomes, so its interpretation is also subject to different methods. If the Rep Test is interpreted formally, it will provide results on the number and range of constructs present within the client’s construct systems. In the hands of an experienced and skilled examiner, information can be gleaned on the equivalence of constructs, thus providing deeper detail on the effective range of the client’s construct system. As more information is obtained from the Rep Test, the better able the therapist will be to guide the therapeutic process (Kelly, 1955a).
Fixed-role therapy is a technique derived from personal construct theory. First, the client prepares a self-characterization sketch , a technique in which the client is asked to write a character sketch about themselves as if they were the principal character in a play, but written as if by a friend who knows the client well. Using information from the self-characterization sketch, as well as from interviews and perhaps the Rep Test, the therapist then writes a fixed-role sketch . The client is asked to act out the fixed-role sketch over a period of weeks. Initially, Kelly and his colleagues emphasized minor changes in the client’s construct systems. However, they later found that it is often easier for a client to play out roles that are the opposite of their usual constructs, rather than making only minor changes in their behavior. Over time, it is expected that the client will learn that the new construct systems are more predictive than their old construct systems, and the fixed-role therapy will establish an ongoing process of reconstruction within the client (Kelly, 1955a).
| Table \(\PageIndex{1}\): The Role Title List Used for the Personal Construct Repertory Test |
|---|
|
discussion question \(\PageIndex{3}\)
Kelly’s fixed-role therapy requires the client to write a script for how they want to live their life. He found that sometimes it was easier for his clients to act out the opposite of their typical behavior. Would you find it easier to make minor changes in your behavior, or easier to make dramatic changes?


11.1.6B: Racial Groups
-
- Last updated
- Save as PDF
- Anonymous
- LibreTexts
The United States is a diverse country, racially and ethnically.
Learning Objectives
- Explain what definitions of race are deployed by the U.S. census
Key Points
- The United States Census Bureau also classifies Americans as ” Hispanic or Latino” and “Not Hispanic or Latino. ” Hispanic and Latino Americans are a racially diverse ethnicity that composes the largest minority group in the nation.
- The one drop rule, a historical colloquial term, stated that any one considered to have even a drop of black blood was to be classified as being black. This was an effort to restore white supremacy during the post Civil War Reconstruction era.
- The Blood Quantum, or Indian Blood, Laws refers to legislation in the United States to establish a person’s membership in Native American tribes or nations.
Key Terms
- One Drop Rule : A historical colloquial term in the United States for the social classification as black of individuals with any African ancestry; meaning any person with “one drop of black blood” was considered black.
- Other Pacific Islander : A United States Census category referring to individuals from the Pacific Islands but not Hawaii.
- ethnicity : The identity of a group of people having common racial, national, religious, or cultural origins.
The United States is a diverse country, racially and ethnically. Six races are officially recognized: white, American Indian and Alaska Native, Asian, black or African American, Native Hawaiian and other Pacific Islander, and people of two or more races. A race called, “Some other race,” is also used in the census and other surveys but is not official.
The United States Census Bureau also classifies Americans as “Hispanic or Latino” and “Not Hispanic or Latino,” which identifies Hispanic and Latino Americans as a racially diverse ethnicity that composes the largest minority group in the nation.
History
The immigrants to the New World of the Americas came largely from ethnically diverse regions of the European Old World. In the Americas, the immigrant populations began to mix among themselves and with the indigenous inhabitants of the continent, as well as the enslaved Africans.
From the beginning of U.S. history, Native Americans, African Americans, and European Americans were classified as belonging to different races. For nearly three centuries, the criteria for membership in these groups were similar, comprising a person’s appearance, their fraction of known non-European ancestry and their social circle. This changed in the late nineteenth century.
Throughout the post-Civil War Reconstruction era, in an effort to restore white supremacy in the South after the emancipation of slaves, the ruling white majority began to classify anyone considered to have “one drop” of “black blood,” or any known African ancestry, to be “black.” In most southern states, this definition was not put into law until the twentieth century. Many local governments established racial segregation of facilities during what came to be known as the Jim Crow era, which began in the late 1800s.
In the twentieth century, efforts to sort the increasingly mixed population of the United States into discrete categories generated many difficulties for the U.S. government (Spickard, 1992). By the standards used in past censuses, many millions of mixed-race children born in the United States have been classified as of a different race than one of their biological parents. Efforts to track mixing between groups led to a proliferation of categories (such as “mulatto” and “octoroon”) and so-called “blood quantum” distinctions, which refers to the degree of ancestry for an individual of a specific racial or ethnic group (e.g., saying someone is “1/4 Omaha tribe”).
These various distinctions became increasingly untethered from self-reported ancestry. Further complicating this fact is that a person’s racial identity can change over time, and self-ascribed race can differ from assigned race (Kressin et al., 2003).
Current Official Definitions of Race and Ethnicity
Aside from their varied social, culture, and political connotations, the idea of racial groups have been used in U.S. censuses as self-identification data items in which residents choose the race or, starting with the 2000 US Census, races with which they most closely identify. Respondents also indicate whether or not they are of Hispanic or Latino origin, which the census considers separately from race. While many see race and ethnicity as the same thing, ethnicity generally refers to a group of people whose members identify with each other through a common heritage and culture, as opposed to the implication of shared biological traits associated with the term “race.”
These categories, therefore, represent a social-political construct for the race or races that respondents consider themselves to be and “generally reflect a social definition of race recognized in this country. ” The concept of race, as outlined for the U.S. Census, has been described as not “scientific or anthropological” and takes into account “social and cultural characteristics as well as ancestry,” using “appropriate scientific methodologies” that are not “primarily biological or genetic in reference. ” The race categories include both racial and national-origin groups.


4.12: Uniformly Integrable Variables
Two of the most important modes of convergence in probability theory are convergence with probability 1 and convergence in mean. As we have noted several times, neither mode of convergence implies the other. However, if we impose an additional condition on the sequence of variables, convergence with probability 1 will imply convergence in mean. The purpose of this brief, but advanced section, is to explore the additional condition that is needed. This section is particularly important for the theory of martingales.
Basic Theory
As usual, our starting point is a random experiment modeled by a probability space \( (\Omega, \mathscr{F}, \P) \). So \( \Omega \) is the set of outcomes, \( \mathscr{F} \) is the \( \sigma \)-algebra of events, and \( \P \) is the probability measure on the sample space \( (\Omega, \mathscr F) \). In this section, all random variables that are mentioned are assumed to be real valued, unless otherwise noted. Next, recall from the section on vector spaces that for \( k \in [1, \infty) \), \( \mathscr{L}_k \) is the vector space of random variables \( X \) with \( \E(|X|^k) \lt \infty \), endowed with the norm \( \|X\|_k = \left[\E(X^k)\right]^{1/k} \). In particular, \( X \in \mathscr{L}_1 \) simply means that \( \E(|X|) \lt \infty \) so that \( \E(X) \) exists as a real number. From the section on expected value as an integral, recall the following notation, assuming of course that the expected value makes sense: \[ \E(X; A) = \E(X \bs{1}_A) = \int_A X \, d\P \]
Definition
The following result is motivation for the main definition in this section.
If \( X \) is a random variable then \( \E(|X|) \lt \infty \) if and only if \( \E(|X|; |X| \ge x) \to 0 \) as \( x \to \infty \).
Proof
Note that that \( |X| \bs{1}(|X| \le x) \) is nonnegative, increasing in \( x \in [0, \infty) \) and \( |X| \bs{1}(|X| \le x) \to |X| \) as \( x \to \infty \). From the monotone convergence theorem, \( \E(|X|; |X| \le x) \to \E(|X|) \) as \( x \to \infty \). On the other hand, \[ \E(|X|) = \E(|X|; |X| \le x) + \E(|X|; |X| \gt x) \] If \( \E(|X|) \lt \infty \) then taking limits in the displayed equation shows that \( \E(|X|: |X| \gt x) \to 0 \) as \( x \to \infty \). On the other hand, \( \E(|X|; |X| \le x) \le x \). So if \( \E(|X|) = \infty \) then \( \E(|X|; |X| \gt x) = \infty \) for every \( x \in [0, \infty) \).
Suppose now that \( X_i \) is a random variable for each \( i \) in a nonempty index set \( I \) (not necessarily countable). The critical definition for this section is to require the convergence in the previous theorem to hold uniformly for the collection of random variables \( \bs X = \{X_i: i \in I\} \).
The collection \( \bs X = \{X_i: i \in I\} \) is uniformly integrable if for each \( \epsilon \gt 0 \) there exists \( x \gt 0 \) such that for all \( i \in I \), \[ \E(|X_i|; |X_i| \gt x) \lt \epsilon \] Equivalently \( \E(|X_i|; |X_i| \gt x) \to 0 \) as \( x \to \infty \) uniformly in \( i \in I \).
Properties
Our next discussion centers on conditions that ensure that the collection of random variables \( \bs X = \{X_i: i \in I\} \) is uniformly integrable. Here is an equivalent characterization:
The collection \( \bs X = \{X_i: i \in I\} \) is uniformly integrable if and only if the following conditions hold:
- \( \{\E(|X_i|): i \in I\} \) is bounded.
- For each \( \epsilon \gt 0 \) there exists \( \delta \gt 0 \) such that if \( A \in \mathscr{F} \) and \( \P(A) \lt \delta \) then \( \E(|X_i|; A) \lt \epsilon \) for all \( i \in I \).
Proof
Suppose that \( \bs X \) is uniformly integrable. With \( \epsilon = 1 \) there exists \( x \gt 0 \) such that \( \E(|X_i|; |X_i| \gt x) \lt 1 \) for all \( i \in I \). Hence \[ \E(|X_i|) = \E(|X_i|; |X_i| \le x) + \E(|X_i|; |X_i| \gt x) \le x + 1, \quad i \in I \] so (a) holds. For (b), let \( \epsilon \gt 0 \). There exists \( x \gt 0 \) such that \( \E(|X_i|; |X_i| \gt x) \lt \epsilon / 2 \) for all \( i \in I \). Let \( \delta = \epsilon / 2 x \). If \( A \in \mathscr{F} \) and \( \P(A) \lt \delta \) then \[ \E(|X_i|; A) = \E(|X_i|; A \cap \{|X| \le x\}) + \E(|X_i|; A \cap \{|X| \gt x\}) \le x \P(A) + \E(|X_i|; |X| \gt x) \lt \epsilon / 2 + \epsilon / 2 = \epsilon\] Conversely, suppose that (a) and (b) hold. By (a), there exists \( c \gt 0 \) such that \( \E(|X_i|) \le c \) for all \( i \in I \). Let \( \epsilon \gt 0 \). By (b) there exists \( \delta \gt 0 \) such that if \( A \in \mathscr{F} \) with \( \P(A) \lt \delta \) then \( \E(|X_i|; A) \lt \epsilon \) for all \( i \in I \). Next, by Markov's inequality, \[ \P(|X_i| \gt x) \le \frac{\E(|X_i|)}{x} \le \frac{c}{x}, \quad i \in I \] Pick \( x \gt 0 \) such that \( c / x \lt \delta \), so that \(\P(|X_i| \gt x) \lt \delta\) for each \( i \in I \). Then for each \( j \in I \), \( \E(|X_i|; |X_j| \gt x) \lt \epsilon \) for all \( i \in I \) and so in particular, \( \E(|X_i|; |X_i| \gt x) \lt \epsilon \) for all \( i \in I \). Hence \( \bs X \) is uniformly integrable.
Condition (a) means that \( \bs X \) is bounded (in norm) as a subset of the vector space \( \mathscr{L}_1 \). Trivially, a finite collection of integrable random variables is uniformly integrable.
Suppose that \( I \) is finite and that \( \E(|X_i|) \lt \infty \) for each \( i \in I \). Then \( \bs X = \{X_i: i \in I\} \) is uniformly integrable.
A subset of a uniformly integrable set of variables is also uniformly integrable.
If \( \{X_i: i \in I\} \) is uniformly integrable and \( J \) is a nonempty subset of \( I \), then \( \{X_j: j \in J\} \) is uniformly integrable.
If the random variables in the collection are dominated in absolute value by a random variable with finite mean, then the collection is uniformly integrable.
Suppose that \( Y \) is a nonnegative random variable with \( \E(Y) \lt \infty \) and that \( |X_i| \le Y \) for each \( i \in I \). Then \( \bs X = \{X_i: i \in I\} \) is uniformly integrable.
Proof
Clearly \( \E(|X_i|; |X_i| \gt x) \le E(Y; Y \gt x) \) for \( x \in [0, \infty) \) and for all \( i \in I \). The right side is independent of \( i \in I \), and by the theorem above , converges to 0 as \( x \to \infty \). Hence \( \bs X \) is uniformly integrable.
The following result is more general, but essentially the same proof works.
Suppose that \( \bs Y = \{X_j: j \in J\} \) is uniformly integrable, and \( \bs X = \{X_i: i \in I\} \) is a set of variables with the property that for each \( i \in I \) there exists \( j \in J \) such that \( |X_i| \le |Y_j| \). Then \( \bs X \) is uniformly integrable.
As a simple corollary, if the variables are bounded in absolute value then the collection is uniformly integrable.
If there exists \( c \gt 0 \) such that \( |X_i| \le c \) for all \( i \in I \) then \( \bs X = \{X_i: i \in I\} \) is uniformly integrable.
Just having \( \E(|X_i|) \) bounded in \( i \in I \) (condition (a) in the characterization above ) is not sufficient for \( \bs X = \{X_i: i \in I\} \) to be uniformly integrable; a counterexample is given below. However, if \( \E(|X_i|^k) \) is bounded in \( i \in I \) for some \( k \gt 1 \), then \( \bs X \) is uniformly integrable. This condition means that \( \bs X \) is bounded (in norm) as a subset of the vector space \( \mathscr{L}_k \).
If \( \left\{\E(|X_i|^k: i \in I\right\} \) is bounded for some \( k \gt 1 \), then \( \{X_i: i \in I\} \) is uniformly integrable.
Proof
Suppose that for some \( k \gt 1 \) and \( c \gt 0 \), \( \E(|X_i|^k) \le c \) for all \( i \in I \). Then \( k - 1 \gt 0 \) and so \( t \mapsto t^{k-1} \) is increasing on \( (0, \infty) \). So if \( |X_i| \gt x \) for \( x \gt 0 \) then \[ |X_i|^k = |X_i| |X_i|^{k-1} \ge |X_i| x^{k-1} \] Hence \( |X_i| \le |X_i|^k / x^{k-1} \) on the event \( |X_i| \gt x \). Therefore \[ \E(|X_i|; |X_i| \gt x) \le \E\left(\frac{|X_i|^k}{x^{k-1}}; |X_i| \gt x\right) \le \frac{\E(|X_i|^k)}{x^{k-1}} \le \frac{c}{x^{k-1}} \] The last expression is independent of \( i \in I \) and converges to 0 as \( x \to \infty \). Hence \( \bs X \) is uniformly integrable.
Uniformly integrability is closed under the operations of addition and scalar multiplication.
Suppose that \( \bs X = \{X_i: i \in I\} \) and \( \bs Y = \{Y_i: i \in I\} \) are uniformly integrable and that \( c \in \R \). Then each of the following collections is also uniformly integrable.
- \( \bs X + \bs Y = \{X_i + Y_i: i \in I\} \)
- \( c \bs X = \{c X_i: i \in I\} \)
Proof
We use the characterization above . The proofs use standard techniques, so try them yourself.
- There exists \( a, \, b \in (0, \infty) \) such that \( \E(|X_i|) \le a \) and \( \E(|Y_i|) \le b \) for all \( i \in I \). Hence \[ \E(|X_i + Y_i|) \le \E(|X_i| + |Y_i|) \le \E(|X_i|) + \E(|Y_i|) \le a + b, \quad i \in I \] Next let \( \epsilon \gt 0 \). There exists \( \delta_1 \gt 0 \) such that if \( A \in \mathscr{F} \) with \( \P(A) \lt \delta_1 \) then \( \E(|X_i|; A) \lt \epsilon / 2 \) for all \( i \in I \), and similarly, there exists \( \delta_2 \gt 0 \) such that if \( A \in \mathscr{F} \) with \( \P(A) \lt \delta_2 \) then \( \E(|Y_i|; A) \lt \epsilon / 2 \) for all \( i \in I \). Hence if \( A \in \mathscr{F} \) with \( \P(A) \lt \delta_1 \wedge \delta_2 \) then \[ \E(|X_i + Y_i|; A) \le \E(|X_i| + |Y_i|; A) = \E(|X_i|; A) + \E(|Y_i|; A) \lt \epsilon / 2 + \epsilon / 2 = \epsilon, \quad i \in I \]
- There exists \( a \in (0, \infty) \) such that \( \E(|X_i|) \le a \) for all \( i \in I \). Hence \[ \E(|c X_i|) = |c| \E(|X_i|) \le c a, \quad i \in I \] The second condition is trivial if \( c = 0 \), so suppose \( c \ne 0 \). For \( \epsilon \gt 0 \) there exists \( \delta \gt 0 \) such that if \( A \in \mathscr{F} \) and \( \P(A) \lt \delta \) then \( \E(|X_i|; A) \lt \epsilon / c \) for all \( i \in I \). Hence \( \E(|c X_i|; A) = |c| \E(|X_i|; A) \lt \epsilon \).
The following corollary is trivial, but will be needed in our discussion of convergence below.
Suppose that \( \{X_i: i \in I\} \) is uniformly integrable and that \( X \) is a random variable with \( \E(|X|) \lt \infty \). Then \( \{X_i - X: i \in I\} \) is uniformly integrable.
Proof
Let \( Y_i = X \) for each \( i \in I \). Then \( \{Y_i: i \in I\} \) is uniformly integrable, so the result follows from the previous theorem.
Convergence
We now come to the main results, and the reason for the definition of uniform integrability in the first place. To set up the notation, suppose that \( X_n \) is a random variable for \( n \in \N_+ \) and that \( X \) is a random variable. We know that if \( X_n \to X \) as \( n \to \infty \) in mean then \( X_n \to X \) as \( n \to \infty \) in probability. The converse is also true if and only if the sequence is uniformly integrable. Here is the first half:
If \( X_n \to X \) as \( n \to \infty \) in mean, then \( \{X_n: n \in \N\} \) is uniformly integrable.
Proof
The hypothesis means that \( X_n \to X \) as \( n \to \infty \) in the vector space \( \mathscr{L}_1 \). That is, \( \E(|X_n|) \lt \infty \) for \( n \in \N_+ \), \( \E(|X|) \lt \infty \), and \( E(|X_n - X|) \to 0 \) as \( n \to \infty \). From the last section, we know that this implies that \( \E(|X_n|) \to \E(|X|) \) as \( n \to \infty \), so \( \E(|X_n|) \) is bounded in \( n \in \N \). Let \( \epsilon \gt 0 \). Then there exists \( N \in \N_+ \) such that if \( n \gt N \) then \( \E(|X_n - X|) \lt \epsilon/2 \). Since all of our variables are in \( \mathscr{L}_1 \), for each \( n \in \N_+ \) there exists \( \delta_n \gt 0 \) such that if \( A \in \mathscr{F} \) and \( \P(A) \lt \delta_n \) then \( \E(|X_n - X|; A) \lt \epsilon / 2 \). Similarly, there exists \( \delta_0 \gt 0 \) such that if \( A \in \mathscr{F} \) and \( \P(A) \lt \delta_0 \) then \( \E(|X|; A) \lt \epsilon / 2 \). Let \( \delta = \min\{\delta_n: n \in \{0, 1, \ldots, N\}\} \) so \( \delta \gt 0 \). If \( A \in \mathscr{F} \) and \( \P(A) \lt \delta \) then \[\E(|X_n|; A) = \E(|X_n - X + X|; A) \le \E(|X_n - X|; A) + \E(|X|; A), \quad n \in \N_+\] If \( n \le N \) then \( \E(|X_n - X|; A) \le \epsilon / 2 \) since \( \delta \le \delta_n \). If \( n \gt N \) then \( \E(|X_n - X|; A) \le \E(|X_n - X|) \lt \epsilon / 2 \). For all \( n \), \( E(|X|; A) \lt \epsilon / 2 \) since \( \delta \le \delta_0 \). So for all \( n \in \N_+ \), \( \E(|X_n|: A) \lt \epsilon \) and hence \( \{X_n: n \in \N_+\} \) is uniformly integrable.
Here is the more important half, known as the uniform integrability theorem :
If \( \{X_n: n \in \N_+\} \) is uniformly integrable and \( X_n \to X \) as \( n \to \infty \) in probability, then \( X_n \to X \) as \( n \to \infty \) in mean.
Proof
Since \( X_n \to X \) as \( n \to \infty \) in probability, we know that there exists a subsequence \( \left(X_{n_k}: k \in \N_+\right) \) of \( (X_n: n \in \N_+) \) such that \( X_{n_k} \to X \) as \( k \to \infty \) with probability 1. By the uniform integrability, \( \E(|X_n|) \) is bounded in \( n \in \N_+ \). Hence by Fatou's lemma \[ \E(|X|) = \E\left(\liminf_{k \to \infty} \left|X_{n_k}\right|\right) \le \liminf_{n \to \infty} \E\left(\left|X_{n_k}\right|\right) \le \limsup_{n \to \infty} \E\left(\left|X_{n_k}\right|\right) \lt \infty \] Let \( Y_n = X_n - X \) for \( n \in \N_+ \). From the corollary above , we know that \( \{Y_n: n \in \N_+\} \) is uniformly integrable, and we also know that \( Y_n \) converges to 0 as \( n \to \infty \) in probability. Hence we need to show that \( Y_n \to 0 \) as \( n \to \infty \) in mean. Let \( \epsilon \gt 0 \). By uniform integrability, there exists \( \delta \gt 0 \) such that if \( A \in \mathscr{F} \) and \( \P(A) \lt \delta \) then \( \E(|Y_n|: A) \lt \epsilon / 2 \) for all \( n \in \N \). Since \( Y_n \to 0 \) as \( n \to \infty \) in probability, there exists \( N \in \N_+ \) such that if \( n \gt N \) then \( \P(|Y_n| \gt \epsilon / 2) \lt \delta \). Hence if \( n \gt N \) then \[ \E(|Y_n|) = \E(|Y_n|; |Y_n| \le \epsilon / 2) + \E(|Y_n|; |Y_n| \gt \epsilon / 2) \lt \epsilon / 2 + \epsilon / 2 = \epsilon \] Hence \( Y_n \to 0 \) as \( n \to \infty \) in mean.
As a corollary, recall that if \( X_n \to X \) as \( n \to \infty \) with probability 1, then \( X_n \to X \) as \( n \to \infty \) in probability. Hence if \( \bs X = \{X_n: n \in \N_+\} \) is uniformly integrable then \( X_n \to X \) as \( n \to \infty \) in mean.
Examples
Our first example shows that bounded \( \mathscr{L}_1 \) norm is not sufficient for uniform integrability.
Suppose that \( U \) is uniformly distributed on the interval \( (0, 1) \) (so \( U \) has the standard uniform distribution ). For \( n \in \N_+ \), let \( X_n = n \bs{1}(U \le 1 / n) \). Then
- \( \E(|X_n|) = 1 \) for all \( n \in \N_+ \)
- \( \E(|X_n|; |X_n| \gt x) = 1 \) for \( x \gt 0 \), \( n \in \N_+ \) with \( n \gt x \)
Proof
First note that \( |X_n| = X_n \) since \( X_n \ge 0 \).
- By definition, \( \E(X_n) = n \P(U \le 1 / n) = n / n = 1 \) for \( n \in \N_+ \).
- If \( n \gt x \gt 0 \) then \( X_n \gt x \) if and only if \( X_n = n \) if and only if \( U \le 1/n \). Hence \( \E(X_n; X_n \gt x) = n \P(U \le 1/n) = 1 \) as before.
By part (b), \( \E(|X_n|; |X_n| \gt x) \) does not converge to 0 as \( x \to \infty \) uniformly in \( n \in \N_+ \), so \( \bs X = \{X_n: n \in \N_+\} \) is not uniformly integrable.
The next example gives an important application to conditional expected value. Recall that if \( X \) is a random variable with \( \E(|X|) \lt \infty \) and \( \mathscr{G} \) is a sub \( \sigma \)-algebra of \( \mathscr{F} \) then \( \E(X \mid \mathscr{G}) \) is the expected value of \( X \) given the information in \( \mathscr{G} \), and is the \( \mathscr{G} \)-measurable random variable closest to \( X \) in a sense. Indeed if \( X \in \mathscr{L}_2(\mathscr{F}) \) then \( \E(X \mid \mathscr{G}) \) is the projection of \( X \) onto \( \mathscr{L}_2(\mathscr{G}) \). The collection of all conditional expected values of \( X \) is uniformly integrable:
Suppose that \( X \) is a real-valued random variable with \( \E(|X|) \lt \infty \). Then \( \{\E(X \mid \mathscr{G}): \mathscr{G} \text{ is a sub }\sigma\text{-algebra of } \mathscr{F}\}\) is uniformly integrable.
Proof
We use the characterization above . Let \( \mathscr{G} \) be a sub \( \sigma \)-algebra of \( \mathscr{F} \). Recall that \( \left|\E(X \mid \mathscr{G})\right| \le \E(|X| \mid \mathscr{G})\) and hence \[ \E[|\E(X \mid \mathscr{G})|] \le \E[\E(|X| \mid \mathscr{G})] = \E(|X|) \] So property (a) holds. Next let \( \epsilon \gt 0 \). Since \( \E(|X|) \lt \infty \), there exists \( \delta \gt 0 \) such that if \( A \in \mathscr{F} \) and \( \P(A) \lt \delta \) then \( \E(|X|; A) \lt \epsilon \). Suppose that \( A \in \mathscr{G} \) with \( \P(A) \lt \delta \). Then \(|\E(X \mid \mathscr{G})| \bs{1}_A \le \E(|X| \mid \mathscr{G}) \bs{1}_A\) so \[ \E[|\E(X \mid \mathscr{G})|; A] \le \E[\E(|X| \mid \mathscr{G}); A] = \E[\E(|X| \bs{1}_A \mid \mathscr{G}] = \E(|X|; A) \lt \epsilon \] So condition (b) holds. Note that the first equality in the displayed equation holds since \( A \in \mathscr{G} \).
Note that the collection of sub \( \sigma \)-algebras of \( \mathscr{F} \), and so also the collection of conditional expected values above, might well be uncountable. The conditional expected values range from \( \E(X) \), when \( \mathscr{G} = \{\Omega, \emptyset\} \) to \( X \) itself, when \( \mathscr{G} = \mathscr{F} \).


 This difficulty can fluctuate depending on where you live, what your family dynamic is like, and a plethora of other factors. Still, even the most “accepting” environments can prove to make queer adolescence more challenging than the average childhood. Personally, I was outed before I was ready. I’d been officially dating a girl for less than a day when my entire school found out about it. Suddenly, I was the pinnacle of queerness.  The token lesbian. As the only “openly” queer kid in a school of 200 students—in a town of 3,000 people—I was expected to have any and all answers to my straight peers’ questions about the LGBT+ world. The only problem? I didn’t have any of the answers myself.
One of the biggest challenges facing the LGBT+ community is a lack of representation in mainstream media. Before I came out, the only gay people I knew about were Ellen Degeneres, Melissa Etheridge, and Elton John. This wasn’t because they were the only queer people in the entertainment industry, but because they were the only ones that society deemed acceptable*. Luckily, I was born at the cusp of the internet boom. While the reliability of Google to teach me everything there was to know about being queer is an entirely separate and frustrating debate; for the time being, we’ll praise my ability to search the web for the answers everyone was asking (that I was asking, too).
I’ve never met a queer person that hasn’t been able to relate to the struggle of trying to find representation in literature. If you think that’s an exaggeration; I challenge you to count the number of movies, TV shows, books, etc. with queer protagonists. If you’re not in the LGBT+ community, you probably thought of Brokeback Mountain, Glee, and not much else. To someone who isn’t queer, it may not seem like a big deal. But can you imagine if there were no movies out there for the heterosexual high-school jock to relate to?
Many of us spend countless hours during our journey towards identification just searching for some sign that we’re not alone in the way we feel. We follow the plot lines of queer characters in shows we don’t watch (I can tell you everything you need to know about the romance between Olivia Wilde and Misha Barton’s characters on the OC ), we sit through horrible acting because it’s better than nothing ( South of Nowhere ), we “ship” same-sex characters that will never be together ( Bend it Like Beckham (2002)), and we watch every single movie under the “Gay and Lesbian” section on Netflix ( Loving Annabelle (2006), Imagine Me & You (2005), If These Walls Could Talk 2 (2000)).
One of my favorite finds during my angsty adolescence search for relatability was The Incredibly True Adventure of 2 Girls in Love (1995).
Enter Walt Whitman—the missing piece to this strangely colloquial and seemingly random blog post. To woo Evie, Randy gifts her a copy of Whitman’s Leaves of Grass (1855). As she reads the lines, “The smoke of my own breath / Echoes, ripples, buzz’d whispers, love-root, silk-thread, crotch and vine,” from “Song of Myself,” Evie asks her sister if she’d ever read Leaves of Grass . Her sister responds, “I had to read it in college. Walt Whitman. He was a homo.” I started reading Whitman after that.
Whitman’s work, like that of many early American authors, continues to be adapted and changed into contemporary forms. While I’m full of critiques about the canon, and I’m certain that Whitman wasn’t the first to write about homoeroticism; his timelessness and influence on literature over the years is something I can’t help but appreciate.
At 14, I was a confused and lonely queer kid growing up with YouTube uploads of bootlegged movies and Google as my source of identity guidance. And, with the help of Whitman, I found some. The idea that Leaves of Grass may have had the same effect on some gay kid growing up in the 19 th century suggests to me that the answer to the question, “Where is American literature?” is “everywhere.”


15.2: Show Me the Money
-
- Last updated
- Save as PDF
1. What is money, what are its characteristics and functions, and what are the three parts of the U.S. money supply?
Money is anything that is acceptable as payment for goods and services. It affects our lives in many ways. We earn it, spend it, save it, invest it—and often wish we had more of it. Businesses and government use money in similar ways. Both require money to finance their operations. By controlling the amount of money in circulation, the federal government can promote economic growth and stability. For this reason, money has been called the lubricant of the machinery that drives our economic system. Our banking system was developed to ease the handling of money.
Characteristics of Money
For money to be a suitable means of exchange, it should have these key characteristics:
- Scarcity: Money should be scarce enough to have some value but not so scarce as to be unavailable. Pebbles, which meet some of the other criteria, would not work well as money because they are widely available. Too much money in circulation increases prices and inflation. Governments control the scarcity of money by limiting the quantity of money in circulation.
- Durability: Any item used as money must be durable.  Even early societies used durable forms of money, such as metal coins and paper money, which lasted for a long time.
- Portability: Money must be easily moved around. Large or bulky items, such as boulders or heavy gold bars, cannot be transported easily from place to place.
- Divisibility: Money must be capable of being divided into smaller parts. Divisible forms of money help make transactions of all sizes and amounts possible.
Table 15.1 provides some interesting facts about our money.
Functions of Money
Using a variety of items as money would be confusing. Thus, societies develop a uniform money system to measure the value of goods and services. For money to be acceptable, it must function as a medium of exchange, as a standard of value, and as a store of value.
As a medium of exchange, money makes transactions easier. Having a common form of payment is much less complicated than having a barter system, wherein goods and services are exchanged for other goods and services. Money allows the exchange of products to be a simple process.
Money also serves as a standard of value. With a form of money whose value is accepted by all, goods and services can be priced in standard units. This makes it easy to measure the value of products and allows transactions to be recorded in consistent terms.
As a store of value, money is used to hold wealth. It retains its value over time, although it may lose some of its purchasing power due to inflation. Individuals may choose to keep their money for future use rather than exchange it today for other types of products or assets.
| Fun Facts about U.S. Currency |
|---|
Did you know . . .
|
Table 15.1 Source: Bureau of Engraving and Printing, “Resources,” https://www.moneyfactory.gov , accessed September 7, 2017.
The U.S. Money Supply
The U.S. money supply is composed of currency, demand deposits, and time deposits. Currency is cash held in the form of coins and paper money. Other forms of currency include travelers’ checks, cashier’s checks, and money orders. The amount of currency in circulation depends on public demand. Domestic demand is influenced primarily by prices for goods and services, income levels, and the availability of alternative payment methods such as credit cards. Until the mid-1980s, nearly all U.S. currency circulated only domestically. Today domestic circulation totals only a small fraction of the total amount of U.S. currency in circulation.
Over the past decade, the amount of U.S. currency has doubled to more than $1.56 trillion and is held both inside and outside the country. 1 Foreign demand is influenced by the political and economic uncertainties associated with some foreign currencies, and recent estimates suggest that between one-half and two-thirds of the value of currency in circulation is held abroad. Some residents of foreign countries hold dollars as a store of value, whereas others use it as a medium of exchange.
Federal Reserve notes make up more than 99 percent of all U.S. currency in circulation. Each year the Federal Reserve Boarddetermines new currency demand and submits a print order to the Treasury’s Bureau of Engraving and Printing (BEP). The order represents the Federal Reserve System’s estimate of the amount of currency the public will need in the upcoming year and reflects estimated changes in currency usage and destruction rates of unfit currency. Table 15.2 shows how long we can expect our money to last on average.
| How Long Will Your Money Last? | |
|---|---|
| Have you ever wondered how quickly money wears out from being handled or damaged? Not surprisingly, smaller denominations have a shorter life span. | |
| $1 bill | 5.8 years |
| $5 bill | 5.5 years |
| $10 bill | 4.5 years |
| $20 bill | 7.9 years |
| $50 bill | 8.5 years |
| $100 bill | 15.0 years |
Table 15.2 Source: “How Long Is the Lifespan of U.S. Paper Money?” https://www.federalreserve.gov , accessed September 7, 2017.
Demand deposits consist of money kept in checking accounts that can be withdrawn by depositors on demand. Demand deposits include regular checking accounts as well as interest-bearing and other special types of checking accounts. Time deposits are deposits at a bank or other financial institution that pay interest but cannot be withdrawn on demand. Examples are certain savings accounts, money market deposit accounts, and certificates of deposit. Economists use two terms to report on and discuss trends in the U.S. monetary system: M1 and M2. M1 (the M stands for money) is used to describe the total amount of readily available money in the system and includes currency and demand deposits. As of August 2017, the M1 monetary supply was $3.5 trillion. M2 includes all M1 monies plus time deposits and other money that is not immediately accessible. In August 2017, the M2 monetary supply was $13.6 trillion. 2 Credit cards, sometimes referred to as “plastic money,” are routinely used as a substitute for cash and checks. Credit cards are not money; they are a form of borrowing. When a bank issues a credit card to a consumer, it gives a short-term loan to the consumer by directly paying the seller for the consumer’s purchases. The consumer pays the credit card company after receiving the monthly statement. Credit cards do not replace money; they simply defer payment.
CONCEPT CHECK
- What is money, and what are its characteristics?
- What are the main functions of money?
- What are the three main components of the U.S. money supply? How do they relate to M1 and M2?


10.9: Assignment- Matched Pairs
The purpose of this activity is to give you guided practice in carrying out the paired t-test and to teach you how to obtain the paired t-test output using statistical software. Here is some background for the historically important data that we are going to work with in this activity.
Background: Gosset’s Seed Plot Data
William S. Gosset was employed by the Guinness brewing company of Dublin. Sample sizes available for experimentation in brewing were necessarily small, and new techniques for handling the resulting data were needed. Gosset consulted Karl Pearson (1857-1936) of University College in London, who told him that the current state of knowledge was unsatisfactory. Gosset undertook a course of study under Pearson, and the outcome of his study was perhaps the most famous paper in statistical literature, “The Probable Error of a Mean” (1908), which introduced the t distribution.
Since Gosset was contractually bound by Guinness, he published under a pseudonym, “Student”; hence, the t distribution is often referred to as Student’s t distribution .
As an example to illustrate his analysis, Gosset reported in his paper on the results of seeding 11 different plots of land with two different types of seed: regular and kiln-dried. There is reason to believe that drying seeds before planting will increase plant yield. Since different plots of soil may be naturally more fertile, this confounding variable was eliminated by using the matched pairs design and planting both types of seed in all 11 plots.
The resulting data (corn yield in pounds per acre) are as follows:
We use these data to test the hypothesis that kiln-dried seed yields more corn than regular seed.
Because of the nature of the experimental design (matched pairs), we are testing the difference in yield.
Note that the differences were calculated: regular – kiln-dried.
Question 1:
State the appropriate hypotheses that are being tested here. Be sure to define the parameter that you are using.
Instructions
Click on the link corresponding to your statistical package to see instructions for completing the activity, and then answer the questions below.
R | StatCrunch | Minitab | Excel | TI Calculator
Question 2:
Are the conditions that allow me to safely use the paired T-test satisfied? Support your answer by using appropriate visual displays.
Question 3:
Based on the visual display that you produced for answering the previous question, does it seem like there is some evidence in the data in favor of the alternative hypothesis? Explain.
Question 4:
Carry out the paired t-test, state the test statistic and P-value, and state your conclusion in context.
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution