24.3: Lipid Metabolism
Learning Objectives
- Explain how energy can be derived from fat
- Explain the purpose and process of ketogenesis
- Describe the process of ketone body oxidation
- Explain the purpose and the process of lipogenesis
Fats (or triglycerides) within the body are ingested as food or synthesized by adipocytes or hepatocytes from carbohydrate precursors (Figure \(\PageIndex{1}\)). Lipid metabolism entails the oxidation of fatty acids to either generate energy or synthesize new lipids from smaller constituent molecules. Lipid metabolism is associated with carbohydrate metabolism, as products of glucose (such as acetyl CoA) can be converted into lipids.
Lipid metabolism begins in the intestine where ingested triglycerides are broken down into smaller chain fatty acids and subsequently into monoglyceride molecules (see Figure \(\PageIndex{1}\).b) by pancreatic lipases , enzymes that break down fats after they are emulsified by bile salts . When food reaches the small intestine in the form of chyme, a digestive hormone called cholecystokinin (CCK) is released by intestinal cells in the intestinal mucosa. CCK stimulates the release of pancreatic lipase from the pancreas and stimulates the contraction of the gallbladder to release stored bile salts into the intestine. CCK also travels to the brain, where it can act as a hunger suppressant.
Together, the pancreatic lipases and bile salts break down triglycerides into free fatty acids. These fatty acids can be transported across the intestinal membrane. However, once they cross the membrane, they are recombined to again form triglyceride molecules. Within the intestinal cells, these triglycerides are packaged along with cholesterol molecules in phospholipid vesicles called chylomicrons (Figure \(\PageIndex{2}\)). The chylomicrons enable fats and cholesterol to move within the aqueous environment of your lymphatic and circulatory systems. Chylomicrons leave the enterocytes by exocytosis and enter the lymphatic system via lacteals in the villi of the intestine. From the lymphatic system, the chylomicrons are transported to the circulatory system. Once in the circulation, they can either go to the liver or be stored in fat cells (adipocytes) that comprise adipose (fat) tissue found throughout the body.
Lipolysis
To obtain energy from fat, triglycerides must first be broken down by hydrolysis into their two principal components, fatty acids and glycerol. This process, called lipolysis , takes place in the cytoplasm. The resulting fatty acids are oxidized by β-oxidation into acetyl CoA, which is used by the Krebs cycle. The glycerol that is released from triglycerides after lipolysis directly enters the glycolysis pathway as DHAP. Because one triglyceride molecule yields three fatty acid molecules with as much as 16 or more carbons in each one, fat molecules yield more energy than carbohydrates and are an important source of energy for the human body. Triglycerides yield more than twice the energy per unit mass when compared to carbohydrates and proteins. Therefore, when glucose levels are low, triglycerides can be converted into acetyl CoA molecules and used to generate ATP through aerobic respiration.
The breakdown of fatty acids, called fatty acid oxidation or beta (β)-oxidation , begins in the cytoplasm, where fatty acids are converted into fatty acyl CoA molecules. This fatty acyl CoA combines with carnitine to create a fatty acyl carnitine molecule, which helps to transport the fatty acid across the mitochondrial membrane. Once inside the mitochondrial matrix, the fatty acyl carnitine molecule is converted back into fatty acyl CoA and then into acetyl CoA (Figure \(\PageIndex{3}\)). The newly formed acetyl CoA enters the Krebs cycle and is used to produce ATP in the same way as acetyl CoA derived from pyruvate.
Ketogenesis
If excessive acetyl CoA is created from the oxidation of fatty acids and the Krebs cycle is overloaded and cannot handle it, the acetyl CoA is diverted to create ketone bodies . These ketone bodies can serve as a fuel source if glucose levels are too low in the body. Ketones serve as fuel in times of prolonged starvation or when patients suffer from uncontrolled diabetes and cannot utilize most of the circulating glucose. In both cases, fat stores are liberated to generate energy through the Krebs cycle and will generate ketone bodies when too much acetyl CoA accumulates.
In this ketone synthesis reaction, excess acetyl CoA is converted into hydroxymethylglutaryl CoA (HMG CoA) . HMG CoA is a precursor of cholesterol and is an intermediate that is subsequently converted into β-hydroxybutyrate, the primary ketone body in the blood (Figure \(\PageIndex{4}\)).
Ketone Body Oxidation
Organs that have classically been thought to be dependent solely on glucose, such as the brain, can actually use ketones as an alternative energy source. This keeps the brain functioning when glucose is limited. When ketones are produced faster than they can be used, they can be broken down into CO 2 and acetone. The acetone is removed by exhalation.  This effect provides one way of telling if a diabetic is properly controlling the disease. The carbon dioxide produced can acidify the blood, leading to diabetic ketoacidosis, a dangerous condition in diabetics.
Ketones oxidize to produce energy for the brain. beta (β)-hydroxybutyrate is oxidized to acetoacetate and NADH is released. An HS-CoA molecule is added to acetoacetate, forming acetoacetyl CoA. The carbon within the acetoacetyl CoA that is not bonded to the CoA then detaches, splitting the molecule in two. This carbon then attaches to another free HS-CoA, resulting in two acetyl CoA molecules. These two acetyl CoA molecules are then processed through the Krebs cycle to generate energy (Figure \(\PageIndex{5}\)).
Lipogenesis
When glucose levels are plentiful, the excess acetyl CoA generated by glycolysis can be converted into fatty acids, triglycerides, cholesterol, steroids, and bile salts. This process, called lipogenesis , creates lipids (fat) from the acetyl CoA and takes place in the cytoplasm of adipocytes (fat cells) and hepatocytes (liver cells). When you eat more glucose or carbohydrates than your body needs, your system uses acetyl CoA to turn the excess into fat. Although there are several metabolic sources of acetyl CoA, it is most commonly derived from glycolysis. Acetyl CoA availability is significant, because it initiates lipogenesis. Lipogenesis begins with acetyl CoA and advances by the subsequent addition of two carbon atoms from another acetyl CoA; this process is repeated until fatty acids are the appropriate length. Because this is a bond-creating anabolic process, ATP is consumed. However, the creation of triglycerides and lipids is an efficient way of storing the energy available in carbohydrates. Triglycerides and lipids, high-energy molecules, are stored in adipose tissue until they are needed.
Although lipogenesis occurs in the cytoplasm, the necessary acetyl CoA is created in the mitochondria and cannot be transported across the mitochondrial membrane. To solve this problem, pyruvate is converted into both oxaloacetate and acetyl CoA. Two different enzymes are required for these conversions. Oxaloacetate forms via the action of pyruvate carboxylase, whereas the action of pyruvate dehydrogenase creates acetyl CoA. Oxaloacetate and acetyl CoA combine to form citrate, which can cross the mitochondrial membrane and enter the cytoplasm. In the cytoplasm, citrate is converted back into oxaloacetate and acetyl CoA. Oxaloacetate is converted into malate and then into pyruvate. Pyruvate crosses back across the mitochondrial membrane to wait for the next cycle of lipogenesis. The acetyl CoA is converted into malonyl CoA that is used to synthesize fatty acids. Figure \(\PageIndex{6}\) summarizes the pathways of lipid metabolism.
Review Questions
Q. Lipids in the diet can be ________.
A. broken down into energy for the body
B. stored as triglycerides for later use
C. converted into acetyl CoA
D. all of the above
Answer: D
Q. The gallbladder provides ________ that aid(s) in transport of lipids across the intestinal membrane.
A. lipases
B. cholesterol
C. proteins
D. bile salts
Answer: D
Q. Triglycerides are transported by chylomicrons because ________.
A. they cannot move easily in the blood stream because they are fat based, while the blood is water based
B. they are too small to move by themselves
C. the chylomicrons contain enzymes they need for anabolism
D. they cannot fit across the intestinal membrane
Answer: A
Q. Which molecule produces the most ATP?
A. carbohydrates
B. FADH 2
C. triglycerides
D. NADH
Answer: C
Q. Which molecules can enter the Krebs cycle?
A. chylomicrons
B. acetyl CoA
C. monoglycerides
D. ketone bodies
Answer: B
Q. Acetyl CoA can be converted to all of the following except ________.
A. ketone bodies
B. fatty acids
C. polysaccharides
D. triglycerides
Answer: C
Critical Thinking Questions
Q. Discuss how carbohydrates can be stored as fat.
A. Carbohydrates are converted into pyruvate during glycolysis. This pyruvate is converted into acetyl CoA and proceeds through the Krebs cycle. When excess acetyl CoA is produced that cannot be processed through the Krebs cycle, the acetyl CoA is converted into triglycerides and fatty acids to be stored in the liver and adipose tissue.
Q. If a diabetic’s breath smells like alcohol, what could this mean?
A. If diabetes is uncontrolled, the glucose in the blood is not being taken up and processed by the cells. Although blood glucose levels are high, there is no glucose available to the cells to be converted into energy. Because glucose is lacking, the body turns to other energy sources, including ketones. A side effect of using ketones as fuel is a sweet alcohol smell on the breath.
Glossary
- beta (β)-hydroxybutyrate
- primary ketone body produced in the body
- beta (β)-oxidation
- fatty acid oxidation
- bile salts
- salts that are released from the liver in response to lipid ingestion and surround the insoluble triglycerides to aid in their conversion to monoglycerides and free fatty acids
- cholecystokinin (CCK)
- hormone that stimulates the release of pancreatic lipase and the contraction of the gallbladder to release bile salts
- chylomicrons
- vesicles containing cholesterol and triglycerides that transport lipids out of the intestinal cells and into the lymphatic and circulatory systems
- fatty acid oxidation
- breakdown of fatty acids into smaller chain fatty acids and acetyl CoA
- hydroxymethylglutaryl CoA (HMG CoA)
- molecule created in the first step of the creation of ketone bodies from acetyl CoA
- ketone bodies
- alternative source of energy when glucose is limited, created when too much acetyl CoA is created during fatty acid oxidation
- lipogenesis
- synthesis of lipids that occurs in the liver or adipose tissues
- lipolysis
- breakdown of triglycerides into glycerol and fatty acids
- monoglyceride molecules
- lipid consisting of a single fatty acid chain attached to a glycerol backbone
- pancreatic lipases
- enzymes released from the pancreas that digest lipids in the diet
- triglycerides
- lipids, or fats, consisting of three fatty acid chains attached to a glycerol backbone


4.3: California State Soil
Many states have a designated state bird, flower, fossil, mineral, etc. In California the state bird is the California Valley Quail, the state flower is the Golden Poppy, the state fossil is the Saber- toothed Cat, and the state mineral is Native Gold. Many states also have a state soil – one that has significance or is important to the state. The San Joaquin is the official state soil of California. Let’s explore how the San Joaquin is important to California and even the entire world.
The Formation of Soil
Weathering is a key part of the process of the soil formation, and the soil is critical to our existence on Earth. In other words, we owe our existence to weathering, and we need to take care of our soil.
Soil forms through the accumulation and decay of organic matter and both mechanical and chemical weathering processes described below. The factors that affect the nature of the soil and the rate of its formation include climate (average temperature and precipitation), the consequent types of vegetation the type of parent rock material, the slope of the surface, and the amount of time available.
Climate
Soils develop because of the weathering of materials on Earth’s surface, including the mechanical breakup of rocks, and the chemical weathering of minerals. Soil development is facilitated by the downward percolation of water. Soil forms most readily under warm temperate to tropical conditions (not cold) and where precipitation amounts are moderate (not dry, but not too wet). Chemical weathering reactions (especially the formation of clay minerals) and biochemical reactions proceed fastest under warm conditions, and plant growth is enhanced in warm climates. Too much water (e.g., in rainforests) can lead to the leaching of important chemical nutrients and hence to acidic soils. In humid and poorly drained regions, swampy conditions may prevail, producing soil that is dominated by organic matter. Too little water (e.g., in deserts and semi-deserts), results in very limited downward chemical transportation and the accumulation of salts and carbonate minerals (e.g., calcite) from upward-moving water. Soils in dry regions also suffer from a lack of organic material.
Parent Material
Soil parent materials can include all different types of bedrock and any type of unconsolidated sediments, such as glacial deposits and stream deposits. Soils are described as residual soils if they develop on bedrock, and transported soils if they develop on transported material such as glacial sediments. But the term “transported soil” is misleading because it implies that the soil itself has been transported, which is not the case. When referring to such soil, it is better to be specific and say, “soil developed on unconsolidated material,” because that distinguishes it from soil developed on bedrock.
Soil Horizons
The process of soil formation generally involves the downward movement of clay, water, and dissolved ions, and a common result of that is the development of chemically and texturally different layers known as soil horizons (figure 4.6). The typically developed soil horizons are:
- O: the layer of organic matter
- A: the layer of partially decayed organic matter mixed with mineral material
- E: the eluviated (leached) layer from which some of the clay and iron have been removed to create a pale layer that may be sandier than the other layers
- B: the layer of accumulation of clay, iron, and other elements from the overlying soil
- C: the layer of incomplete weathering
- R: the parent material or bedrock
Another type of layer that develops in hot arid regions (such as in the Mojave Desert) is known as caliche (pronounced ca-lee-chee ). It forms from the downward (or in some cases upward) movement of calcium ions, and the precipitation of calcite within the soil. Well-developed caliche may cement the surrounding material together to form a layer that has the consistency of concrete.
Like all geological materials, the soil is subject to erosion, although, under natural conditions on gentle slopes, the rate of soil formation either balances or exceeds the rate of erosion. Human practices related to home building, forestry and agriculture have significantly upset this balance.
Soils are held in place by vegetation. When vegetation is removed, either through cutting trees or routinely harvesting crops and tilling the soil, that protection is either temporarily or permanently lost. The primary agents of the erosion of unprotected soil are water and wind.
Case Study - The San Joaquin Series
The San Joaquin soil was initially documented and officially established in California in 1900 and therefore is the oldest, continuously recognized soil series within the state. The process of establishing the San Joaquin as the official state soil in California began with the Professional Soil Scientists Association of California (PSSAC).
Inspired by a visit to the USDA offices in Madera and Hanford, California, science teacher Alex Lehman envisioned a project to connect students and the public with soil conservation. This project, titled "Proposing a California State Soil—Preserving a Legacy and a Commitment to Future Generations," sparked a unique learning experience for students at Martin Luther King, Jr. Middle School in Madera.
Students embraced a cross-disciplinary approach, integrating science, math, English, social studies, and history with conservation education. Social studies classes delved into the lawmaking process, English classes saw students crafting proposals and poems about soil, science and art classes collaborated on science fair displays, and music classes even produced an official state soil song.
The San Joaquin Soil Series became the official state soil of California on August 20, 1997. It was the culmination of the efforts by students and teachers, natural resources professionals, the PSSAC, state legislators, and various state universities.
What is the San Joaquin Soil?
Every soil can be separated into three separate particle size fractions called sand, silt, and clay, which makes up the soil texture. In addition, the arrangement of soil horizons can tell you a lot about the age of a soil and how it developed over time. Some common horizon designations are A, E, B, and C or combinations of those.
Typically, San Joaquin soils have a brown to reddish brown surface or surface soil horizon with a loam (a combination of sand, silt, and clay) texture that has an accumulation of organic matter. The next underlying horizon is similar in soil texture to the surface horizons; however, it does not have an accumulation of organic matter. Iron oxides released in this horizon are not masked so much by the organic matter and impart a redder hue to the soil when moist. This horizon rests on a brown or reddish-brown clay or clay loam horizon that has a distinctive prismatic structure (soil broken up into pillar like structures) with cracks between the prisms. This horizon, with a dramatic increase in clay which restricts root and water penetration, is about 15 to 30 inches 38 to 76 cm (38 to 76 cm) from the surface of the soil.
The first characteristic in a San Joaquin soil that people notice is a distinctive soil horizon (layer) known as a duripan to soil scientists and generally as “hardpan” to many people. It is extremely hard, and it can be chipped with mechanical means or through use of a pick or very strong, heavy shovel. However, there is much more to the San Joaquin soil than just the duripan.
Finally, the fourth horizon is the brown to reddish brown silica cemented duripan. The duripan has an abrupt upper boundary at a depth of 50 to 101 cm (20 to 40 inches) and is impervious to roots and water. The duripan often continues to a depth of 152 cm (60 inches) or more.


16.8: Nucleophilic Aromatic Substitution
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Objectives
After completing this section, you should be able to
- identify the conditions necessary for an aryl halide to undergo nucleophilic aromatic substitution, and give an example of such a reaction.
- write the detailed mechanism for a nucleophilic aromatic substitution reaction.
- compare the mechanism of a nucleophilic aromatic substitution reaction and the S N 1 and S N 2 mechanisms discussed earlier.
- identify the product formed when a given nucleophile reacts with a given aryl halide in a nucleophilic aromatic substitution reaction.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- Meisenheimer complex
- nucleophilic aromatic substitution
Study Notes
A nucleophilic aromatic substitution reaction is a reaction in which one of the substituents in an aromatic ring is replaced by a nucleophile.
A Meisenheimer complex is a negatively charged intermediate formed by the attack of a nucleophile upon one of the aromatic-ring carbons during the course of a nucleophilic aromatic substitution reaction. A typical Meisenheimer complex is shown in the reaction scheme below. Notice how this particular complex can be formed from two different starting materials by using a different nucleophile in each case.
A Nucleophilic Aromatic Displacement Reactions of Aryl Halides
The carbon-halogen bonds of aryl halides are like those of alkenyl halides in being much stronger than those of alkyl halides. The simple aryl halides generally are resistant to attack by nucleophiles in either S N 1 or S N 2 reactions. However, this low reactivity can be changed dramatically by changes in the reaction conditions and the structure of the aryl halide. In fact, nucleophilic displacement becomes quite rapid
- when the aryl halide is activated by substitution with strongly electron-attracting groups such as NO 2 , and
- when very strongly basic nucleophilic reagents are used.
Addition-Elimination Mechanism of Nucleophilic Substitution of Aryl Halides
Although the simple aryl halides are inert to the usual nucleophilic reagents, considerable activation is produced by strongly electron-attracting substituents provided these are located in either the ortho or para positions, or both. For example, the displacement of chloride ion from 1-chloro-2,4-dinitrobenzene by dimethylamine occurs readily in ethanol solution at room temperature. Under the same conditions chlorobenzene completely fails to react; thus the activating influence of the two nitro groups amounts to a factor of at least 108:
A related reaction is that of 2,4-dinitrofluorobenzene with the amino groups of peptides and proteins, and this reaction provides a means for analysis of the N-terminal amino acids in polypeptide chains.
In general, the reactions of activated aryl halides closely resemble the S N 2-displacement reactions of aliphatic halides. The same nucleophilic reagents are effective (e.g., CH 3 O ⊖ , HO ⊖ , and RNH 2 ); the reactions are second order overall (first order in halide and first order in nucleophile); and for a given halide the more nucleophilic the attacking reagent, the faster the reaction. However, there must be more than a subtle difference in mechanism because an aryl halide is unable, to pass through the same type of transition state as an alkyl halide in S N 2 displacements.
The generally accepted mechanism of nucleophilic aromatic substitution of aryl halides carrying activating groups involves two steps. The first step involves attack of the nucleophile Y: ⊖ at the carbon bearing the halogen substituent to form an intermediate carbanion. The aromatic system is destroyed on forming the anion, and the carbon at the reaction site changes from planar (sp 2 bonds) to tetrahedral (sp 3 bonds).
In the second step, loss of an anion, X ⊖ or Y ⊖ , regenerates an aromatic system, and, if X ⊖ is lost, the overall reaction is nucleophilic displacement of X by Y.
In the case of a neutral nucleophilic reagent, Y or HY, the reaction sequence would be the same except for the necessary adjustments in the charge of the intermediate:
Why is this reaction pathway generally unfavorable for the simple aryl halides? The answer is that the intermediate is too high in energy to be formed at any practical rate. Not only has it lost the aromatic stabilization of the benzene ring, but its formation results in transfer of a negative charge to the ring carbons, which themselves are not very electronegative.
However, when strongly electron-attracting groups are located on the ring at the ortho-para positions, the intermediate anion is stabilized by delocalization of electrons from the ring carbons to more favorable locations on the substituent groups. As an example, consider the displacement of bromine by methoxid (OCH 3 ) in the reaction of 4-bromonitrobenzene and methoxide ion:
The anionic intermediate formed by addition of methoxide ion to the aryl halide can be described by the valence-bond structures 5a-5d. Of these structures 5d is especially important because in it the charge is transferred from the ring carbons to the oxygen of the nitro substituent:
Substituents in the meta positions have much less effect on the reactivity of an aryl halide because delocalization of electrons to the substituent is not possible. No formulas can be written analogous to 5c and 5d in which the negative charges are both on atoms next to positive nitrogen, and
In a few instances, stable compounds resembling the postulated reaction intermediate have been isolated. One classic example is the complex 7 (isolated by J. Meisenheimer), which is the product of the reaction of either the methyl aryl ether 6 with potassium ethoxide, or the ethyl aryl ether 8 and potassium methoxide:
Exercises
Exercise \(\PageIndex{1}\)
Propose a mechanism for the following reaction:
- Answer
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5: Intercultural and International Communication
Chapter Objectives
The purpose of this chapter is to:
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Define high-context and low-context cultures
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Explain the differences between globalization and global reach
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Understand how business writing can bridge cultural differences
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Define culture and intercultural communication
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Define monochromic and polychronic time factors
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Understand how business writing can foster community peacebuilding
WET Design Company and Its Corporate Culture
Employees at WET Design ("Wet and Wild") created the Fountains of Bellagio in Las Vegas, Nevada. What business communication significance does the Fountains of Bellagio hold? The answer is collaboration and innovation. Mark W. Fuller is the founder and CEO of WET Design. He has implemented a collaborative corporate culture model in which every employee is immersed in the company’s structure.
When employees begin employment at WET, they spend two weeks working in all departments and experiencing a variety of tasks in each department. Throughout their tenure, employees are offered training opportunities pertaining to the company, in addition to creative classes on a variety of subjects such as meditation, woodworking, comedy improvisation, and fitness. Mr. Fuller believes this structure sparks creativity, innovation, collaboration, communication, and versatility, which are characteristics of strong teams. The goal is to give clients the best product and to give employees the opportunity for professional and personal development and growth.
Source: Jamila Kibibi-Kendall for Kibibi Creations, CC BY
Types of Cultural Communication
Culture is a learned system of behaviors and meanings that is shared amongst groups within a family, a community, or a nation. It is a set of common values, ideas, and actions that are passed from one generation to the next. Culture is multifaceted; therefore, it influences communication. It requires thinking beyond tolerance to learning about and embracing cultural differences.
Diversity recognizes individual differences and lived experiences. Subculture represents the ideas and ways of life of people within a diverse group. The range of human differences include but not limited to:
- Religious and spiritual beliefs
- Race
- Ancestry
- Nationality or ethnic origin
- Sex or gender
- Gender identity and/or gender expression
- Sexual orientation
- Ability
- Health
- Age and generation
- Military or veteran status
- Socioeconomic status and background
- Personal life experience
- Learning style and personality
The lived experiences of individuals in a subculture may differ from individuals within a larger culture.
Inclusive Language
Business communities and stakeholders are diverse. To ensure your writing promotes inclusion, justice, and equity, it is essential to use nondiscriminatory language.
Gender identification is an individual's personal choice. Stakeholders may not fit traditional gender categories; therefore, to presume gender is an act of exclusion. In business writing it is acceptable to use the plural pronoun they/their as a singular nongendered pronoun.
Example:
Each associate uploaded the performance training questionnaire to their evaluation folder.
You may add your preferred gender pronouns to your signature block closing.
Example:
Sincerely,
Valerie A. Gray, Professor of English
Pronouns: She/Her/Hers
When you are communicating the same message to several receivers, for example a letter to employees regarding additional health care coverage, use plural pronouns or plural titles (associates, coordinators, facilitators, supervisors, trainers, etc).
Example:
All coordinators were invited to the debriefing meeting.
To learn more about gender cultural aspects and the history of inclusive pronouns, visit the University of Wisconsin at Milwaukee Lesbian, Gay, Bisexual, Transgender, Queer Plus (LGBTQ+) Resource Center
Ethical Consideration
It is important to learn about cultures and subcultures. However, it is equally as important to avoid stereotyping individuals within a culture and subculture. A stereotype is an oversimplified opinion of a group and applying that opinion to everyone in that group. Stereotyping can lead to prejudice, bias, racism, and discrimination.
Understanding International Communication
International communication , also referred to as global communication , is the verbal, nonverbal, and oral communication that occurs between individuals from different international locations.
Source: Photo by Wonderlane on Unsplash
Global reach refers to the ability of a company to have customers worldwide.
Globalization refers to a company from one country agreeing to conduct business with a company in another country.
This video explains the complexities of globalization and its impact on various segments in business. As you view the video, consider the ways in which it is easy to hinder written business communication in global situations.
Ethical Considerations for Cultural Global Advantage
In order to understand why businesses are willing to operate in a complex global environment, it is important to understand two fundamental concepts that drive almost all business decisions: absolute and comparative advantage. Countries and companies are willing to assume the risk of engaging in global trade because they believe that they have an advantage over the competition that they can turn into profits. Not all countries have the same natural resources, infrastructure, labor force, or technology. These differences create advantages that can be exploited in global trade, to a country’s (or company’s) benefit. As you analyze the audience and explain advantages and benefits, it is necessary to acknowledge concerns that citizens of the country have regarding their community in order to avoid damage to it.
Absolute Advantage
An entity (country, region, company, or individual) is considered to have an absolute advantage if either of the following conditions exists:
- It is the only source of a particular product, good, or service. This kind of absolute advantage is very rare and usually depends on a particular natural resource being available only within a certain region or country.
- An entity is also considered to have an absolute advantage if it is able to produce more of something than another entity while using the same amount of resources (factors of production).
Comparative Advantage
An entity (country, region, company, or individual) is considered to have a comparative advantage over another in producing a particular good or service if it can produce the good or service at a lower relative opportunity cost. (The section Ethical Considerations for Cultural Global Advantage is a derivative of Absolute and Comparative Advantage . Authored by : Linda Williams. Provided by : Lumen Learning. License : CC BY: Attribution )
The following videos explain the role globalization plays in international businesses.
Cultural Competence
This video will assistance you with understanding how the formation of cultural competency will improve your international communication skills.
Understanding Intercultural Communication
Intercultural communication is the ability to communicate sensitively, authenticity, and clearly with people from other cultures and countries. In order to be successful in cultural communication, intercultural awareness is a necessity. Anthropologist Edward T. Hall’s cultural awareness led to the development of the types of culture: high-context and low-context.
In high-context cultures , interpersonal nonverbal communication is the primary way in which meaning is interpreted. Interpersonal nonverbal communication is expressed through eye contact, gestures, and the time it takes to respond to business correspondence. Because of their ability to read nonverbal cues, communicators in high-context cultures expect communicators from low-context cultures to interpret nonverbal communication; therefore, high-context communicators are more reserved in their speech. Hierarchial leadership is the business communication model.
In low-context cultures , verbal communication is the primary way in which meaning is interpreted. Verbal communication is expressed openly through discussing a topic face-to-face and reading detailed business documents.
Business-Specific Cultural Differences
Collectivism and Individualism
Within a business, collectivism is reflected in the team or collaborative approach; the performance of the group has precedence over each individual in the group. The preferred structure to complete tasks for high-context cultures is through collectivism.
Within a business, individualism is reflected in the individual approach; the performance of the individuals within the group has precedence. Individuals are expected to contribute to the group process. The preferred structure to complete tasks for low-context cultures is through individualism. Group members are often encouraged to share their opinions.
Knowledge is based on what is already known about the topic, in order to understand context and meaning. Communicators read nonverbal behavior and the environment in which the communication takes place. The following table provides additional categories of intercultural nonverbal communication.
| Nonverbal Communication | Cultural Context |
| Body language | Checking messages on your phone, fidgeting, or having a closed body position may suggest that you are uninterested or uncomfortable. In turn, your audience may feel disrespected or believe such actions are unprofessional. |
| Business versus personal relationships | Discussing your interests, family, and hobbies is an effective way to break the ice and build rapport with counterparts from other cultures. |
| Clothing | Clothing communicates cultural norms, trends, style, tradition, religion, status, and comfort. |
| Expressions and Eye Contact | Nodding and maintaining eye contact may indicate agreement, but not always. To ensure that everyone agrees and/or understands, incorporate various communication modes, written and oral, to ensure everyone agrees. |
| Gestures | Gestures are hand or head movements used to signal information such as emotions, attitudes, and intentions. The use and interpretation of gestures are based on one’s culture. Therefore, gestures are not universal. |
| Space | Social distancing is a term that is in the international lexicon. Personal space throughout the world is six feet. Before COVID-19, it was customary for citizens of high-context cultures to communicate in close proximity of three feet or less. |
| Time | Time is categorized as monochronic and polychronic. Monochronic refers to doing or focusing on one task. Polychronic refers to doing or focusing on multiple tasks , or multitasking.  When a meeting is scheduled to begin at 1:00 p.m. they expect the meeting to begin on time.  When a meeting is scheduled for 1:00 p.m., they may not attend until 1:15 p.m. The lateness is not necessary disrespectful. Instead, polychronic individual prefer to take a leisurely and reflective approach to business. |
High Context and Low-Context Spectrum of Countries: A Representative Table
The following chart categorizes high-context and low-context cultures.
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High Context |
Low Context |
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Brazil |
Canada |
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China |
Denmark |
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Colombia |
Finland |
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Ghana |
Germany |
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Mexico |
Norway |
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Saudi Arabia |
United States |
High-Context and Low-Context Business Communication Differences: A Representative Table
The following chart presents differences in business communication based on high-context and low-context cultures.
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High Context |
Low Context |
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An oral agreement finalizes a deal |
A written contract finalizes a deal |
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Complements are deflected |
Complements are embraced |
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Word choice and style are formal |
Business communication is formal or semi-formal, depending on the medium and the audience |
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Face-to-face communication relies more on nonverbal cues |
Face-to-face communication relies more on oral cues |
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Time is spent discussing family, friends, and hobbies |
Time is spent discussing business |
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Tasks are performed one at a time |
Multitasking is common |
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Negative information is not communicated bluntly in order to avoid embarrassment for the recipient (also known as “saving face”); an indirect communication method is preferred |
Negative information is communicated at times, too bluntly; a direct communication method is preferred |
The following videos provide an overview of nonverbal intercultural communication. The videos are sponsored by the U.S. Department of State.
Formalize Your Writing to Produce Clarity in Intercultural Communication
To lessen miscommunication between cultures and within cultures, focus on explaining meaning to ensure readers understand your points. Your writing goal is denotation and eliminating abstract expressions such as clichés, stock expressions. Denotation refers to using words that are concrete and explanations that are clear to ensure that all members in your audience have the same understanding.
Another way to clarify your writing is to eliminate euphemisms. Euphemisms refer to abstract expressions that are used within a culture or subculture as a way to soften a word or phrase that may be too harsh.
Nuances of Language
Readers may interpret written text based on the language and grammar conventions of their primary language. It may be difficult to switch to the standards of another language because of the variances in word choice, structure, development, grammar, and style. Additional time to complete the drafting and editing stages of the writing process, and collaborating with colleagues may be necessary.
Peacebuilding, Justice, and Business Communication
Peacebuilding is a key factor to ensuring a company does not take advantage of a country's resources. The term peacebuilding ("United Nations," para.1) was first introduced by Johan Galtung, a conflict and peace scholar, to present strategies for lasting peace by addressing the “root causes” of violent conflict. Peacebuilding addresses the reasons for conflict and establishes ways for peaceful relationships through empowerment by those in conflict to reach peaceful solutions. In this way, peacebuilding serves as a business strategy for all stakeholders to work together to create a culture of peace. Peacebuilding in business encompasses employers and employees providing a safe work environment free from harassment and bullying; governance and policing that is fair and just; safe environmental conditions regardless of location; and a global perspective that incorporates cultural differences.
Stakeholders can practice peacebuilding on a small scale by, for example:
- Marching for a social justice cause
- Boycotting a company that does not share your view on a social justice issue
- Writing to the CEO of a company to complain about its hiring policies
- Signing petitions and emailing the CEO to encourage her or him to offer only non-genetically modified foods
These categories require companies to have an environmental and social agenda:
- Safe workplace
- Fair policies
- Safe environment
- Global perspective
Cultural Considerations Summary
This chapter provides strategies for writing documents that address international and intercultural communication, and the responsibilities stakeholders assign to businesses that are local and international. To lessen miscommunication between cultures and within cultures, focus on writing with denotation and eliminating abstract expressions such as clichés, euphemisms, and expressions.
General Intercultural Writing Tips
The successful intercultural business communicator is:
• Aware that their preferred values and behaviors are influenced by culture and are not necessarily correct
• Aware that language choices can lead to miscommunication
• Flexible and open to change
• Sensitive to verbal and nonverbal behavior
• Sensitive to differences among individuals within a culture
• Aware of the values, beliefs, and practices in other cultures
• Aware that one way of conducting business is not the only way
• Celebrates commonalities and differences
All links live as of July 2021.
References
Ramberg, J.J. (reporter). (2014, April 27). “Wet and Wild: Creating A Culture Of Versatility.” Your Money. MSNBC. Retrieved from https://www.youtube.com/watch?v=zQf4uNeyGUo
United Nations Peacebuilding Support Office. (n.d.). Peacebuilding & The United Nations. Retrieved from http://www.un.org/en/peacebuilding/pbso/pbun.shtml
This work “Intercultural and International Business Communication” is licensed under CC BY by Valerie A. Gray


7.2: The Method of Moments
Basic Theory
The Method
Suppose that we have a basic random experiment with an observable, real-valued random variable \(X\). The distribution of \(X\) has \(k\) unknown real-valued parameters, or equivalently, a parameter vector \(\bs{\theta} = (\theta_1, \theta_2, \ldots, \theta_k)\) taking values in a parameter space, a subset of \( \R^k \). As usual, we repeat the experiment \(n\) times to generate a random sample of size \(n\) from the distribution of \(X\). \[ \bs{X} = (X_1, X_2, \ldots, X_n) \] Thus, \(\bs{X}\) is a sequence of independent random variables, each with the distribution of \(X\). The method of moments is a technique for constructing estimators of the parameters that is based on matching the sample moments with the corresponding distribution moments . First, let \[ \mu^{(j)}(\bs{\theta}) = \E\left(X^j\right), \quad j \in \N_+ \] so that \(\mu^{(j)}(\bs{\theta})\) is the \(j\)th moment of \(X\) about 0. Note that we are emphasizing the dependence of these moments on the vector of parameters \(\bs{\theta}\). Note also that \(\mu^{(1)}(\bs{\theta})\) is just the mean of \(X\), which we usually denote simply by \(\mu\). Next, let \[ M^{(j)}(\bs{X}) = \frac{1}{n} \sum_{i=1}^n X_i^j, \quad j \in \N_+ \] so that \(M^{(j)}(\bs{X})\) is the \(j\)th sample moment about 0. Equivalently, \(M^{(j)}(\bs{X})\) is the sample mean for the random sample \(\left(X_1^j, X_2^j, \ldots, X_n^j\right)\) from the distribution of \(X^j\). Note that we are emphasizing the dependence of the sample moments on the sample \(\bs{X}\). Note also that \(M^{(1)}(\bs{X})\) is just the ordinary sample mean, which we usually just denote by \(M\) (or by \( M_n \) if we wish to emphasize the dependence on the sample size). From our previous work, we know that \(M^{(j)}(\bs{X})\) is an unbiased and consistent estimator of \(\mu^{(j)}(\bs{\theta})\) for each \(j\). Here's how the method works:
To construct the method of moments estimators \(\left(W_1, W_2, \ldots, W_k\right)\) for the parameters \((\theta_1, \theta_2, \ldots, \theta_k)\) respectively, we consider the equations \[ \mu^{(j)}(W_1, W_2, \ldots, W_k) = M^{(j)}(X_1, X_2, \ldots, X_n) \] consecutively for \( j \in \N_+ \) until we are able to solve for \(\left(W_1, W_2, \ldots, W_k\right)\) in terms of \(\left(M^{(1)}, M^{(2)}, \ldots\right)\).
The equations for \( j \in \{1, 2, \ldots, k\} \) give \(k\) equations in \(k\) unknowns, so there is hope (but no guarantee) that the equations can be solved for \( (W_1, W_2, \ldots, W_k) \) in terms of \( (M^{(1)}, M^{(2)}, \ldots, M^{(k)}) \). In fact, sometimes we need equations with \( j \gt k \). Exercise 28 below gives a simple example. The method of moments can be extended to parameters associated with bivariate or more general multivariate distributions, by matching sample product moments with the corresponding distribution product moments. The method of moments also sometimes makes sense when the sample variables \( (X_1, X_2, \ldots, X_n) \) are not independent, but at least are identically distributed. The hypergeometric model below is an example of this.
Of course, the method of moments estimators depend on the sample size \( n \in \N_+ \). We have suppressed this so far, to keep the notation simple. But in the applications below, we put the notation back in because we want to discuss asymptotic behavior.
Estimates for the Mean and Variance
Estimating the mean and variance of a distribution are the simplest applications of the method of moments. Throughout this subsection, we assume that we have a basic real-valued random variable \( X \) with \( \mu = \E(X) \in \R \) and \( \sigma^2 = \var(X) \in (0, \infty) \). Occasionally we will also need \( \sigma_4 = \E[(X - \mu)^4] \), the fourth central moment. We sample from the distribution of \( X \) to produce a sequence \( \bs X = (X_1, X_2, \ldots) \) of independent variables, each with the distribution of \( X \). For each \( n \in \N_+ \), \( \bs X_n = (X_1, X_2, \ldots, X_n) \) is a random sample of size \( n \) from the distribution of \( X \). We start by estimating the mean, which is essentially trivial by this method.
Suppose that the mean \(\mu\) is unknown. The method of moments estimator of \( \mu \) based on \( \bs X_n \) is the sample mean \[ M_n = \frac{1}{n} \sum_{i=1}^n X_i\]
- \( \E(M_n) = \mu \) so \( M_n \) is unbiased for \( n \in \N_+ \)
- \( \var(M_n) = \sigma^2/n \) for \( n \in \N_+ \)so \( \bs M = (M_1, M_2, \ldots) \) is consistent.
Proof
It does not get any more basic than this. The method of moments works by matching the distribution mean with the sample mean. The fact that \( \E(M_n) = \mu \) and \( \var(M_n) = \sigma^2 / n \) for \( n \in \N_+ \) are properties that we have seen several times before.
Estimating the variance of the distribution, on the other hand, depends on whether the distribution mean \( \mu \) is known or unknown. First we will consider the more realistic case when the mean in also unknown. Recall that for \( n \in \{2, 3, \ldots\} \), the sample variance based on \( \bs X_n \) is \[ S_n^2 = \frac{1}{n - 1} \sum_{i=1}^n (X_i - M_n)^2 \] Recall also that \(\E(S_n^2) = \sigma^2\) so \( S_n^2 \) is unbiased for \( n \in \{2, 3, \ldots\} \), and that \(\var(S_n^2) = \frac{1}{n} \left(\sigma_4 - \frac{n - 3}{n - 1} \sigma^4 \right)\) so \( \bs S^2 = (S_2^2, S_3^2, \ldots) \) is consistent.
Suppose that the mean \( \mu \) and the variance \( \sigma^2 \) are both unknown. For \( n \in \N_+ \), the method of moments estimator of \(\sigma^2\) based on \( \bs X_n \) is \[T_n^2 = \frac{1}{n} \sum_{i=1}^n (X_i - M_n)^2\]
- \(\bias(T_n^2) = -\sigma^2 / n\) for \( n \in \N_+ \) so \( \bs T^2 = (T_1^2, T_2^2, \ldots) \) is asymptotically unbiased.
- \(\mse(T_n^2) = \frac{1}{n^3}\left[(n - 1)^2 \sigma_4 - (n^2 - 5 n + 3) \sigma^4\right]\) for \( n \in \N_+ \) so \( \bs T^2 \) is consistent.
Proof
As before, the method of moments estimator of the distribution mean \(\mu\) is the sample mean \(M_n\). On the other hand, \(\sigma^2 = \mu^{(2)} - \mu^2\) and hence the method of moments estimator of \(\sigma^2\) is \(T_n^2 = M_n^{(2)} - M_n^2\), which simplifies to the result above. Note that \(T_n^2 = \frac{n - 1}{n} S_n^2\) for \( n \in \{2, 3, \ldots\} \).
- Note that \(\E(T_n^2) = \frac{n - 1}{n} \E(S_n^2) = \frac{n - 1}{n} \sigma^2\), so \(\bias(T_n^2) = \frac{n-1}{n}\sigma^2 - \sigma^2 = -\frac{1}{n} \sigma^2\).
- Recall that \(\mse(T_n^2) = \var(T_n^2) + \bias^2(T_n^2)\). But \(\var(T_n^2) = \left(\frac{n-1}{n}\right)^2 \var(S_n^2)\). The result follows from substituting \(\var(S_n^2)\) given above and \(\bias(T_n^2)\) in part (a).
Hence \( T_n^2 \) is negatively biased and on average underestimates \(\sigma^2\). Because of this result, \( T_n^2 \) is referred to as the biased sample variance to distinguish it from the ordinary (unbiased) sample variance \( S_n^2 \).
Next let's consider the usually unrealistic (but mathematically interesting) case where the mean is known, but not the variance.
Suppose that the mean \( \mu \) is known and the variance \( \sigma^2 \) unknown. For \( n \in \N_+ \), the method of moments estimator of \(\sigma^2\) based on \( \bs X_n \) is \[ W_n^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2 \]
- \( \E(W_n^2) = \sigma^2 \) so \( W_n^2 \) is unbiased for \( n \in \N_+ \)
- \(\var(W_n^2) = \frac{1}{n}(\sigma_4 - \sigma^4)\) for \( n \in \N_+ \) so \( \bs W^2 = (W_1^2, W_2^2, \ldots) \) is consistent.
Proof
These results follow since \( \W_n^2 \) is the sample mean corresponding to a random sample of size \( n \) from the distribution of \( (X - \mu)^2 \).
We compared the sequence of estimators \( \bs S^2 \) with the sequence of estimators \( \bs W^2 \) in the introductory section on Estimators. Recall that \( \var(W_n^2) \lt \var(S_n^2) \) for \( n \in \{2, 3, \ldots\} \) but \( \var(S_n^2) / \var(W_n^2) \to 1 \) as \( n \to \infty \). There is no simple, general relationship between \( \mse(T_n^2) \) and \( \mse(S_n^2) \) or between \( \mse(T_n^2) \) and \( \mse(W_n^2) \), but the asymptotic relationship is simple.
\( \mse(T_n^2) / \mse(W_n^2) \to 1 \) and \( \mse(T_n^2) / \mse(S_n^2) \to 1 \) as \( n \to \infty \)
Proof
In light of the previous remarks, we just have to prove one of these limits. The first limit is simple, since the coefficients of \( \sigma_4 \) and \( \sigma^4 \) in \( \mse(T_n^2) \) are asymptotically \( 1 / n \) as \( n \to \infty \).
It also follows that if both \( \mu \) and \( \sigma^2 \) are unknown, then the method of moments estimator of the standard deviation \( \sigma \) is \( T = \sqrt{T^2} \). In the unlikely event that \( \mu \) is known, but \( \sigma^2 \) unknown, then the method of moments estimator of \( \sigma \) is \( W = \sqrt{W^2} \).
Estimating Two Parameters
There are several important special distributions with two paraemters; some of these are included in the computational exercises below. With two parameters, we can derive the method of moments estimators by matching the distribution mean and variance with the sample mean and variance, rather than matching the distribution mean and second moment with the sample mean and second moment. This alternative approach sometimes leads to easier equations. To setup the notation, suppose that a distribution on \( \R \) has parameters \( a \) and \( b \). We sample from the distribution to produce a sequence of independent variables \( \bs X = (X_1, X_2, \ldots) \), each with the common distribution. For \( n \in \N_+ \), \( \bs X_n = (X_1, X_2, \ldots, X_n) \) is a random sample of size \( n \) from the distribution. Let \( M_n \), \( M_n^{(2)} \), and \( T_n^2 \) denote the sample mean, second-order sample mean, and biased sample variance corresponding to \( \bs X_n \), and let \( \mu(a, b) \), \( \mu^{(2)}(a, b) \), and \( \sigma^2(a, b) \) denote the mean, second-order mean, and variance of the distribution.
If the method of moments estimators \( U_n \) and \( V_n \) of \( a \) and \( b \), respectively, can be found by solving the first two equations \[ \mu(U_n, V_n) = M_n, \quad \mu^{(2)}(U_n, V_n) = M_n^{(2)} \] then \( U_n \) and \( V_n \) can also be found by solving the equations \[ \mu(U_n, V_n) = M_n, \quad \sigma^2(U_n, V_n) = T_n^2 \]
Proof
Recall that \( \sigma^2(a, b) = \mu^{(2)}(a, b) - \mu^2(a, b) \). In addition, \( T_n^2 = M_n^{(2)} - M_n^2 \). Hence the equations \( \mu(U_n, V_n) = M_n \), \( \sigma^2(U_n, V_n) = T_n^2 \) are equivalent to the equations \( \mu(U_n, V_n) = M_n \), \( \mu^{(2)}(U_n, V_n) = M_n^{(2)} \).
Because of this result, the biased sample variance \( T_n^2 \) will appear in many of the estimation problems for special distributions that we consider below.
Special Distributions
The Normal Distribution
The normal distribution with mean \( \mu \in \R \) and variance \( \sigma^2 \in (0, \infty) \) is a continuous distribution on \( \R \) with probability density function \( g \) given by \[ g(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp\left[-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2\right], \quad x \in \R \] This is one of the most important distributions in probability and statistics, primarily because of the central limit theorem. The normal distribution is studied in more detail in the chapter on Special Distributions.
Suppose now that \( \bs{X} = (X_1, X_2, \ldots, X_n) \) is a random sample of size \( n \) from the normal distribution with mean \( \mu \) and variance \( \sigma^2 \). Form our general work above , we know that if \( \mu \) is unknown then the sample mean \( M \) is the method of moments estimator of \( \mu \), and if in addition, \( \sigma^2 \) is unknown then the method of moments estimator of \( \sigma^2 \) is \( T^2 \). On the other hand, in the unlikely event that \( \mu \) is known then \( W^2 \) is the method of moments estimator of \( \sigma^2 \). Our goal is to see how the comparisons above simplify for the normal distribution.
Mean square errors of \( S_n^2 \) and \( T_n^2 \).
- \(\mse(T^2) = \frac{2 n - 1}{n^2} \sigma^4\)
- \(\mse(S^2) = \frac{2}{n - 1} \sigma^4\)
- \(\mse(T^2) \lt \mse(S^2)\) for \(n \in \{2, 3, \ldots, \}\)
Proof
Recall that for the normal distribution, \(\sigma_4 = 3 \sigma^4\). Substituting this into the general results gives parts (a) and (b). Part (c) follows from (a) and (b). Of course the asymptotic relative efficiency is still 1, from our previous theorem.
Thus, \(S^2\) and \(T^2\) are multiplies of one another; \(S^2\) is unbiased, but when the sampling distribution is normal, \(T^2\) has smaller mean square error. Surprisingly, \(T^2\) has smaller mean square error even than \(W^2\).
Mean square errors of \( T^2 \) and \( W^2 \).
- \(\mse(W^2) = \frac{2}{n} \sigma^4\)
- \(\mse(T^2) \lt \mse(W^2)\) for \(n \in \{2, 3, \ldots\}\)
Proof
Again, since the sampling distribution is normal, \(\sigma_4 = 3 \sigma^4\). Substituting this into the gneral formula for \(\var(W_n^2)\) gives part (a).
Run the normal estimation experiment 1000 times for several values of the sample size \(n\) and the parameters \(\mu\) and \(\sigma\). Compare the empirical bias and mean square error of \(S^2\) and of \(T^2\) to their theoretical values. Which estimator is better in terms of bias? Which estimator is better in terms of mean square error?
Next we consider estimators of the standard deviation \( \sigma \). As noted in the general discussion above, \( T = \sqrt{T^2} \) is the method of moments estimator when \( \mu \) is unknown, while \( W = \sqrt{W^2} \) is the method of moments estimator in the unlikely event that \( \mu \) is known. Another natural estimator, of course, is \( S = \sqrt{S^2} \), the usual sample standard deviation. The following sequence, defined in terms of the gamma function turns out to be important in the analysis of all three estimators.
Consider the sequence \[ a_n = \sqrt{\frac{2}{n}} \frac{\Gamma[(n + 1) / 2)}{\Gamma(n / 2)}, \quad n \in \N_+ \] Then \( 0 \lt a_n \lt 1 \) for \( n \in \N_+ \) and \( a_n \uparrow 1 \) as \( n \uparrow \infty \).
First, assume that \( \mu \) is known so that \( W_n \) is the method of moments estimator of \( \sigma \).
For \( n \in \N_+ \),
- \( \E(W) = a_n \sigma \)
- \( \bias(W) = (a_n - 1) \sigma \)
- \( \var(W) = \left(1 - a_n^2\right) \sigma^2 \)
- \( \mse(W) = 2 (1 - a_n) \sigma^2 \)
Proof
Recall that \(U^2 = n W^2 / \sigma^2 \) has the chi-square distribution with \( n \) degrees of freedom, and hence \( U \) has the chi distribution with \( n \) degrees of freedom. Solving gives \[ W = \frac{\sigma}{\sqrt{n}} U \] From the formulas for the mean and variance of the chi distribution we have \begin{align*} \E(W) & = \frac{\sigma}{\sqrt{n}} \E(U) = \frac{\sigma}{\sqrt{n}} \sqrt{2} \frac{\Gamma[(n + 1) / 2)}{\Gamma(n / 2)} = \sigma a_n \\ \var(W) & = \frac{\sigma^2}{n} \var(U) = \frac{\sigma^2}{n}\left\{n - [\E(U)]^2\right\} = \sigma^2\left(1 - a_n^2\right) \end{align*}
Thus \( W \) is negatively biased as an estimator of \( \sigma \) but asymptotically unbiased and consistent. Of course we know that in general (regardless of the underlying distribution), \( W^2 \) is an unbiased estimator of \( \sigma^2 \) and so \( W \) is negatively biased as an estimator of \( \sigma \). In the normal case, since \( a_n \) involves no unknown parameters, the statistic \( W / a_n \) is an unbiased estimator of \( \sigma \). Next we consider the usual sample standard deviation \( S \).
For \( n \in \{2, 3, \ldots\} \),
- \( \E(S) = a_{n-1} \sigma \)
- \( \bias(S) = (a_{n-1} - 1) \sigma \)
- \( \var(S) = \left(1 - a_{n-1}^2\right) \sigma^2 \)
- \( \mse(S) = 2 (1 - a_{n-1}) \sigma^2 \)
Proof
Recall that \(V^2 = (n - 1) S^2 / \sigma^2 \) has the chi-square distribution with \( n - 1 \) degrees of freedom, and hence \( V \) has the chi distribution with \( n - 1 \) degrees of freedom. The proof now proceeds just as in the previous theorem, but with \( n - 1 \) replacing \( n \).
As with \( W \), the statistic \( S \) is negatively biased as an estimator of \( \sigma \) but asymptotically unbiased, and also consistent. Since \( a_{n - 1}\) involves no unknown parameters, the statistic \( S / a_{n-1} \) is an unbiased estimator of \( \sigma \). Note also that, in terms of bias and mean square error, \( S \) with sample size \( n \) behaves like \( W \) with sample size \( n - 1 \). Finally we consider \( T \), the method of moments estimator of \( \sigma \) when \( \mu \) is unknown.
For \( n \in \{2, 3, \ldots\} \),
- \( \E(T) = \sqrt{\frac{n - 1}{n}} a_{n-1} \sigma \)
- \( \bias(T) = \left(\sqrt{\frac{n - 1}{n}} a_{n-1} - 1\right) \sigma \)
- \( \var(T) = \frac{n - 1}{n} \left(1 - a_{n-1}^2 \right) \sigma^2 \)
- \( \mse(T) = \left(2 - \frac{1}{n} - 2 \sqrt{\frac{n-1}{n}} a_{n-1} \right) \sigma^2 \)
Proof
The results follow easily from the previous theorem since \( T_n = \sqrt{\frac{n - 1}{n}} S_n \).
The Bernoulli Distribution
Recall that an indicator variable is a random variable \( X \) that takes only the values 0 and 1. The distribution of \( X \) is known as the Bernoulli distribution , named for Jacob Bernoulli, and has probability density function \( g \) given by \[ g(x) = p^x (1 - p)^{1 - x}, \quad x \in \{0, 1\} \] where \( p \in (0, 1) \) is the success parameter . The mean of the distribution is \( p \) and the variance is \( p (1 - p) \).
Suppose now that \( \bs{X} = (X_1, X_2, \ldots, X_n) \) is a random sample of size \( n \) from the Bernoulli distribution with unknown success parameter \( p \). Since the mean of the distribution is \( p \), it follows from our general work above that the method of moments estimator of \( p \) is \( M \), the sample mean. In this case, the sample \( \bs{X} \) is a sequence of Bernoulli trials, and \( M \) has a scaled version of the binomial distribution with parameters \( n \) and \( p \): \[ \P\left(M = \frac{k}{n}\right) = \binom{n}{k} p^k (1 - p)^{n - k}, \quad k \in \{0, 1, \ldots, n\} \] Note that since \( X^k = X \) for every \( k \in \N_+ \), it follows that \( \mu^{(k)} = p \) and \( M^{(k)} = M \) for every \( k \in \N_+ \). So any of the method of moments equations would lead to the sample mean \( M \) as the estimator of \( p \). Although very simple, this is an important application, since Bernoulli trials are found embedded in all sorts of estimation problems, such as empirical probability density functions and empirical distribution functions.
The Geometric Distribution
The geometric distribution on \(\N_+\) with success parameter \(p \in (0, 1)\) has probability density function \( g \) given by \[ g(x) = p (1 - p)^{x-1}, \quad x \in \N_+ \] The geometric distribution on \( \N_+ \) governs the number of trials needed to get the first success in a sequence of Bernoulli trials with success parameter \( p \). The mean of the distribution is \(\mu = 1 / p\).
Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the geometric distribution on \( \N_+ \) with unknown success parameter \(p\). The method of moments estimator of \(p\) is \[U = \frac{1}{M}\]
Proof
The method of moments equation for \(U\) is \(1 / U = M\).
The geometric distribution on \( \N \) with success parameter \( p \in (0, 1) \) has probability density function \[ g(x) = p (1 - p)^x, \quad x \in \N \] This version of the geometric distribution governs the number of failures before the first success in a sequence of Bernoulli trials. The mean of the distribution is \( \mu = (1 - p) \big/ p \).
Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the geometric distribution on \( \N \) with unknown parameter \(p\). The method of moments estimator of \(p\) is \[U = \frac{1}{M + 1}\]
Proof
The method of moments equation for \(U\) is \((1 - U) \big/ U = M\).
The Negative Binomial Distribution
More generally, the negative binomial distribution on \( \N \) with shape parameter \( k \in (0, \infty) \) and success parameter \( p \in (0, 1) \) has probability density function \[ g(x) = \binom{x + k - 1}{k - 1} p^k (1 - p)^x, \quad x \in \N \] If \( k \) is a positive integer, then this distribution governs the number of failures before the \( k \)th success in a sequence of Bernoulli trials with success parameter \( p \). However, the distribution makes sense for general \( k \in (0, \infty) \). The negative binomial distribution is studied in more detail in the chapter on Bernoulli Trials. The mean of the distribution is \( k (1 - p) \big/ p \) and the variance is \( k (1 - p) \big/ p^2 \). Suppose now that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the negative binomial distribution on \( \N \) with shape parameter \( k \) and success parameter \( p \)
If \( k \) and \( p \) are unknown, then the corresponding method of moments estimators \( U \) and \( V \) are \[ U = \frac{M^2}{T^2 - M}, \quad V = \frac{M}{T^2} \]
Proof
Matching the distribution mean and variance to the sample mean and variance gives the equations \[ U \frac{1 - V}{V} = M, \quad U \frac{1 - V}{V^2} = T^2 \]
As usual, the results are nicer when one of the parameters is known.
Suppose that \( k \) is known but \( p \) is unknown. The method of moments estimator \( V_k \) of \( p \) is \[ V_k = \frac{k}{M + k} \]
Proof
Matching the distribution mean to the sample mean gives the equation \[ k \frac{1 - V_k}{V_k} = M \]
Suppose that \( k \) is unknown but \( p \) is known. The method of moments estimator of \( k \) is \[ U_p = \frac{p}{1 - p} M \]
- \( \E(U_p) = k \) so \( U_p \) is unbiased.
- \( \var(U_p) = \frac{k}{n (1 - p)} \) so \( U_p \) is consistent.
Proof
Matching the distribution mean to the sample mean gives the equation \( U_p \frac{1 - p}{p} = M\).
- \( E(U_p) = \frac{p}{1 - p} \E(M)\) and \(\E(M) = \frac{1 - p}{p} k\)
- \( \var(U_p) = \left(\frac{p}{1 - p}\right)^2 \var(M) \) and \( \var(M) = \frac{1}{n} \var(X) = \frac{1 - p}{n p^2} \)
The Poisson Distribution
The
Poisson distribution
with parameter \( r \in (0, \infty) \) is a discrete distribution on \( \N \) with probability density function \( g \) given by \[ g(x) = e^{-r} \frac{r^x}{x!}, \quad x \in \N \] The mean and variance are both \( r \). The distribution is named for Simeon Poisson and is widely used to model the number of
random points
is a region of time or space. The parameter \( r \) is proportional to the size of the region, with the proportionality constant playing the role of the average
rate
at which the points are distributed in time or space. The Poisson distribution is studied in more detail in the chapter on the Poisson Process.
Suppose now that \( \bs{X} = (X_1, X_2, \ldots, X_n) \) is a random sample of size \( n \) from the Poisson distribution with parameter \( r \). Since \( r \) is the mean, it follows from our general work above that the method of moments estimator of \( r \) is the sample mean \( M \).
The Gamma Distribution
The gamma distribution with shape parameter \(k \in (0, \infty) \) and scale parameter \(b \in (0, \infty)\) is a continuous distribution on \( (0, \infty) \) with probability density function \( g \) given by \[ g(x) = \frac{1}{\Gamma(k) b^k} x^{k-1} e^{-x / b}, \quad x \in (0, \infty) \] The gamma probability density function has a variety of shapes, and so this distribution is used to model various types of positive random variables. The gamma distribution is studied in more detail in the chapter on Special Distributions. The mean is \(\mu = k b\) and the variance is \(\sigma^2 = k b^2\).
Suppose now that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample from the gamma distribution with shape parameter \(k\) and scale parameter \(b\).
Suppose that \(k\) and \(b\) are both unknown, and let \(U\) and \(V\) be the corresponding method of moments estimators. Then \[ U = \frac{M^2}{T^2}, \quad V = \frac{T^2}{M}\]
Proof
Matching the distribution mean and variance with the sample mean and variance leads to the equations \(U V = M\), \(U V^2 = T^2\). Solving gives the results.
The method of moments estimators of \(k\) and \(b\) given in the previous exercise are complicated, nonlinear functions of the sample mean \(M\) and the sample variance \(T^2\). Thus, computing the bias and mean square errors of these estimators are difficult problems that we will not attempt. However, we can judge the quality of the estimators empirically, through simulations.
When one of the parameters is known, the method of moments estimator of the other parameter is much simpler.
Suppose that \(k\) is unknown, but \(b\) is known. The method of moments estimator of \( k \) is \[U_b = \frac{M}{b}\]
- \( \E(U_b) = k \) so \(U_b\) is unbiased.
- \(\var(U_b) = k / n\) so \(U_b\) is consistent.
Proof
If \(b\) is known, then the method of moments equation for \(U_b\) is \(b U_b = M\). Solving gives (a). Next, \(\E(U_b) = \E(M) / b = k b / b = k\), so \(U_b\) is unbiased. Finally \(\var(U_b) = \var(M) / b^2 = k b ^2 / (n b^2) = k / n\).
Suppose that \(b\) is unknown, but \(k\) is known. The method of moments estimator of \(b\) is \[V_k = \frac{M}{k}\]
- \( \E(V_k) = b \) so \(V_k\) is unbiased.
- \( \var(V_k) = b^2 / k n \) so that \(V_k\) is consistent.
Proof
If \(k\) is known, then the method of moments equation for \(V_k\) is \(k V_k = M\). Solving gives (a). Next, \(\E(V_k) = \E(M) / k = k b / k = b\), so \(V_k\) is unbiased. Finally \(\var(V_k) = \var(M) / k^2 = k b ^2 / (n k^2) = b^2 / k n\).
Run the gamma estimation experiment 1000 times for several different values of the sample size \(n\) and the parameters \(k\) and \(b\). Note the empirical bias and mean square error of the estimators \(U\), \(V\), \(U_b\), and \(V_k\). One would think that the estimators when one of the parameters is known should work better than the corresponding estimators when both parameters are unknown; but investigate this question empirically.
The Beta Distribution
The beta distribution with left parameter \(a \in (0, \infty) \) and right parameter \(b \in (0, \infty)\) is a continuous distribution on \( (0, 1) \) with probability density function \( g \) given by \[ g(x) = \frac{1}{B(a, b)} x^{a-1} (1 - x)^{b-1}, \quad 0 \lt x \lt 1 \] The beta probability density function has a variety of shapes, and so this distribution is widely used to model various types of random variables that take values in bounded intervals. The beta distribution is studied in more detail in the chapter on Special Distributions. The first two moments are \(\mu = \frac{a}{a + b}\) and \(\mu^{(2)} = \frac{a (a + 1)}{(a + b)(a + b + 1)}\).
Suppose now that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the beta distribution with left parameter \(a\) and right parameter \(b\).
Suppose that \(a\) and \(b\) are both unknown, and let \(U\) and \(V\) be the corresponding method of moments estimators. Then \[U = \frac{M \left(M - M^{(2)}\right)}{M^{(2)} - M^2}, \quad V = \frac{(1 - M)\left(M - M^{(2)}\right)}{M^{(2)} - M^2}\]
Proof
The method of moments equations for \(U\) and \(V\) are \[\frac{U}{U + V} = M, \quad \frac{U(U + 1)}{(U + V)(U + V + 1)} = M^{(2)}\] Solving gives the result.
The method of moments estimators of \(a\) and \(b\) given in the previous exercise are complicated nonlinear functions of the sample moments \(M\) and \(M^{(2)}\). Thus, we will not attempt to determine the bias and mean square errors analytically, but you will have an opportunity to explore them empricially through a simulation.
Suppose that \(a\) is unknown, but \(b\) is known. Let \(U_b\) be the method of moments estimator of \(a\). Then \[ U_b = b \frac{M}{1 - M} \]
Proof
If \(b\) is known then the method of moments equation for \(U_b\) as an estimator of \(a\) is \(U_b \big/ (U_b + b) = M\). Solving for \(U_b\) gives the result.
Suppose that \(b\) is unknown, but \(a\) is known. Let \(V_a\) be the method of moments estimator of \(b\). Then \[ V_a = a \frac{1 - M}{M} \]
Proof
If \(a\) is known then the method of moments equation for \(V_a\) as an estimator of \(b\) is \(a \big/ (a + V_a) = M\). Solving for \(V_a\) gives the result.
Run the beta estimation experiment 1000 times for several different values of the sample size \(n\) and the parameters \(a\) and \(b\). Note the empirical bias and mean square error of the estimators \(U\), \(V\), \(U_b\), and \(V_a\). One would think that the estimators when one of the parameters is known should work better than the corresponding estimators when both parameters are unknown; but investigate this question empirically.
The following problem gives a distribution with just one parameter but the second moment equation from the method of moments is needed to derive an estimator.
Suppose that \( \bs{X} = (X_1, X_2, \ldots, X_n) \) is a random sample from the symmetric beta distribution, in which the left and right parameters are equal to an unknown value \( c \in (0, \infty) \). The method of moments estimator of \( c \) is \[ U = \frac{2 M^{(2)}}{1 - 4 M^{(2)}} \]
Proof
Note that the mean \( \mu \) of the symmetric distribution is \( \frac{1}{2} \), independently of \( c \), and so the first equation in the method of moments is useless. However, matching the second distribution moment to the second sample moment leads to the equation \[ \frac{U + 1}{2 (2 U + 1)} = M^{(2)} \] Solving gives the result.
The Pareto Distribution
The Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) is a continuous distribution on \( (b, \infty) \) with probability density function \( g \) given by \[ g(x) = \frac{a b^a}{x^{a + 1}}, \quad b \le x \lt \infty \] The Pareto distribution is named for Vilfredo Pareto and is a highly skewed and heavy-tailed distribution. It is often used to model income and certain other types of positive random variables. The Pareto distribution is studied in more detail in the chapter on Special Distributions. If \(a \gt 2\), the first two moments of the Pareto distribution are \(\mu = \frac{a b}{a - 1}\) and \(\mu^{(2)} = \frac{a b^2}{a - 2}\).
Suppose now that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the Pareto distribution with shape parameter \(a \gt 2\) and scale parameter \(b \gt 0\).
Suppose that \(a\) and \(b\) are both unknown, and let \(U\) and \(V\) be the corresponding method of moments estimators. Then \begin{align} U & = 1 + \sqrt{\frac{M^{(2)}}{M^{(2)} - M^2}} \\ V & = \frac{M^{(2)}}{M} \left( 1 - \sqrt{\frac{M^{(2)} - M^2}{M^{(2)}}} \right) \end{align}
Proof
The method of moments equations for \(U\) and \(V\) are \begin{align} \frac{U V}{U - 1} & = M \\ \frac{U V^2}{U - 2} & = M^{(2)} \end{align} Solving for \(U\) and \(V\) gives the results.
As with our previous examples, the method of moments estimators are complicatd nonlinear functions of \(M\) and \(M^{(2)}\), so computing the bias and mean square error of the estimator is difficult. Instead, we can investigate the bias and mean square error empirically, through a simulation.
Run the Pareto estimation experiment 1000 times for several different values of the sample size \(n\) and the parameters \(a\) and \(b\). Note the empirical bias and mean square error of the estimators \(U\) and \(V\).
When one of the parameters is known, the method of moments estimator for the other parameter is simpler.
Suppose that \(a\) is unknown, but \(b\) is known. Let \(U_b\) be the method of moments estimator of \(a\). Then \[ U_b = \frac{M}{M - b}\]
Proof
If \(b\) is known then the method of moment equation for \(U_b\) as an estimator of \(a\) is \(b U_b \big/ (U_b - 1) = M\). Solving for \(U_b\) gives the result.
Suppose that \(b\) is unknown, but \(a\) is known. Let \(V_a\) be the method of moments estimator of \(b\). Then \[V_a = \frac{a - 1}{a}M\]
- \( \E(V_a) = b \) so \(V_a\) is unbiased.
- \(\var(V_a) = \frac{b^2}{n a (a - 2)}\) so \(V_a\) is consistent.
Proof
If \(a\) is known then the method of moments equation for \(V_a\) as an estimator of \(b\) is \(a V_a \big/ (a - 1) = M\). Solving for \(V_a\) gives (a). Next, \(\E(V_a) = \frac{a - 1}{a} \E(M) = \frac{a - 1}{a} \frac{a b}{a - 1} = b\) so \(V_a\) is unbiased. Finally, \(\var(V_a) = \left(\frac{a - 1}{a}\right)^2 \var(M) = \frac{(a - 1)^2}{a^2} \frac{a b^2}{n (a - 1)^2 (a - 2)} = \frac{b^2}{n a (a - 2)}\).
The Uniform Distribution
The (continuous)
uniform distribution
with location parameter \( a \in \R \) and scale parameter \( h \in (0, \infty) \) has probability density function \( g \) given by \[ g(x) = \frac{1}{h}, \quad x \in [a, a + h] \] The distribution models a point chosen
at random
from the interval \( [a, a + h] \). The mean of the distribution is \( \mu = a + \frac{1}{2} h \) and the variance is \( \sigma^2 = \frac{1}{12} h^2 \). The uniform distribution is studied in more detail in the chapter on Special Distributions. Suppose now that \( \bs{X} = (X_1, X_2, \ldots, X_n) \) is a random sample of size \( n \) from the uniform distribution.
Suppose that \( a \) and \( h \) are both unknown, and let \( U \) and \( V \) denote the corresponding method of moments estimators. Then \[ U = 2 M - \sqrt{3} T, \quad V = 2 \sqrt{3} T \]
Proof
Matching the distribution mean and variance to the sample mean and variance leads to the equations \( U + \frac{1}{2} V = M \) and \( \frac{1}{12} V^2 = T^2 \). Solving gives the result.
As usual, we get nicer results when one of the parameters is known.
Suppose that \( a \) is known and \( h \) is unknown, and let \( V_a \) denote the method of moments estimator of \( h \). Then \[ V_a = 2 (M - a) \]
- \( \E(V_a) = h \) so \( V \) is unbiased.
- \( \var(V_a) = \frac{h^2}{3 n} \) so \( V_a \) is consistent.
Proof
Matching the distribution mean to the sample mean leads to the equation \( a + \frac{1}{2} V_a = M \). Solving gives the result.
- \( \E(V_a) = 2[\E(M) - a] = 2(a + h/2 - a) = h \)
- \( \var(V_a) = 4 \var(M) = \frac{h^2}{3 n} \)
Suppose that \( h \) is known and \( a \) is unknown, and let \( U_h \) denote the method of moments estimator of \( a \). Then \[ U_h = M - \frac{1}{2} h \]
- \( \E(U_h) = a \) so \( U_h \) is unbiased.
- \( \var(U_h) = \frac{h^2}{12 n} \) so \( U_h \) is consistent.
Proof
Matching the distribution mean to the sample mean leads to the quation \( U_h + \frac{1}{2} h = M \). Solving gives the result.
- \( \E(U_h) = \E(M) - \frac{1}{2}h = a + \frac{1}{2} h - \frac{1}{2} h = a \)
- \( \var(U_h) = \var(M) = \frac{h^2}{12 n} \)
The Hypergeometric Model
Our basic assumption in the method of moments is that the sequence of observed random variables \( \bs{X} = (X_1, X_2, \ldots, X_n) \) is a random sample from a distribution. However, the method makes sense, at least in some cases, when the variables are identically distributed but dependent. In the hypergeometric model , we have a population of \( N \) objects with \( r \) of the objects type 1 and the remaining \( N - r \) objects type 0 . The parameter \( N \), the population size , is a positive integer. The parameter \( r \), the type 1 size , is a nonnegative integer with \( r \le N \). These are the basic parameters, and typically one or both is unknown. Here are some typical examples:
- The objects are devices, classified as good or defective .
- The objects are persons, classified as female or male .
- The objects are voters, classified as for or against a particular candidate.
- The objects are wildlife or a particular type, either tagged or untagged .
We sample \( n \) objects from the population at random, without replacement. Let \( X_i \) be the type of the \( i \)th object selected, so that our sequence of observed variables is \( \bs{X} = (X_1, X_2, \ldots, X_n) \). The variables are identically distributed indicator variables, with \( P(X_i = 1) = r / N \) for each \( i \in \{1, 2, \ldots, n\} \), but are dependent since the sampling is without replacement. The number of type 1 objects in the sample is \( Y = \sum_{i=1}^n X_i \). This statistic has the hypergeometric distribution with parameter \( N \), \( r \), and \( n \), and has probability density function given by \[ P(Y = y) = \frac{\binom{r}{y} \binom{N - r}{n - y}}{\binom{N}{n}} = \binom{n}{y} \frac{r^{(y)} (N - r)^{(n - y)}}{N^{(n)}}, \quad y \in \{\max\{0, N - n + r\}, \ldots, \min\{n, r\}\} \] The hypergeometric model is studied in more detail in the chapter on Finite Sampling Models.
As above, let \( \bs{X} = (X_1, X_2, \ldots, X_n) \) be the observed variables in the hypergeometric model with parameters \( N \) and \( r \). Then
- The method of moments estimator of \( p = r / N \) is \( M = Y / n \), the sample mean.
- The method of moments estimator of \( r \) with \( N \) known is \( U = N M = N Y / n \).
- The method of moments estimator of \( N \) with \( r \) known is \( V = r / M = r n / Y \) if \( Y > 0 \).
Proof
These results all follow simply from the fact that \( \E(X) = \P(X = 1) = r / N \).
In the voter example (3) above, typically \( N \) and \( r \) are both unknown, but we would only be interested in estimating the ratio \( p = r / N \). In the reliability example (1), we might typically know \( N \) and would be interested in estimating \( r \). In the wildlife example (4), we would typically know \( r \) and would be interested in estimating \( N \). This example is known as the capture-recapture model.
Clearly there is a close relationship between the hypergeometric model and the Bernoulli trials model above. In fact, if the sampling is with replacement, the Bernoulli trials model would apply rather than the hypergeometric model. In addition, if the population size \( N \) is large compared to the sample size \( n \), the hypergeometric model is well approximated by the Bernoulli trials model.


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5.1
Algebraic Expressions
-
Algebra is useful because it allows us to understand many situations in real life by modeling them with expressions.
-
Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
-
Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
-
Algebraic expressions are often simplified and evaluated using the four arithmetic operations.
5.2
Linear Equations in One Variable with Applications
-
Solving linear equations means discovering what the value of the variable in a linear equation represents in the given conditions.
-
When solving a linear equation, most often you will have one solution; however, a linear equation may have no solutions or infinitely many solutions.
5.3
Linear Inequalities in One Variable with Applications
-
Inequalities can be used when the possible values (answers) in a certain situation are numerous, or when the exact value (answer) is not known, but it is known to be within a range of possible values.
-
Linear inequalities can be represented using a number line or using interval notation.
5.4
Ratios and Proportions
-
A ratio is a comparison of two numbers. The ratio of two numbers
and
can be written as:
to
OR
:
OR the fraction
/
.
-
All fractions are ratios, but not all ratios are fractions. Ratios make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only.
-
When two ratios are equal, we say they are in proportion or are proportional.
-
Setting up proportions allows us to solve many various situations where three of the four values of the proportion are known.
5.5
Graphing Linear Equations and Inequalities
-
Linear equations can be represented graphically on a rectangular coordinate system.
-
Solving linear equations in two variables means finding the point where two lines intersect. There are three possibilities: The lines intersect at exactly one point; the lines do not intersect (they are parallel); or the lines intersect everywhere (they are the same line).
-
Solving linear inequalities in two variables means finding a region of possible answers. Every point in this region will make both inequalities true statements.
-
Plotting points is a standard way to help graph linear equations and linear inequalities.
5.6
Quadratic Equations with Two Variables with Applications
-
A quadratic equation is an algebraic equation where the highest power (degree) of the equation is two.
-
To solve a quadratic equation is to find the value(s) that when substituted in for the variables, will make the equation equal to zero.
-
There can be two, one, or no solutions to any quadratic equation.
-
There are several methods to solve a quadratic equation. These methods include factoring quadratic equations, graphic quadratic equations, using the square root method, and using the quadratic formula.
5.7
Functions
-
A relation is any set of ordered pairs
. All of the \(x\)-values of the set are the domain, and all of the
-values of the set are the range.
-
A relation is a function if each \(x\)-value in the domain is assigned to exactly one element in the range. A
-value in the range can have more than one \(x\)-value assigned to it; but each \(x\)-value can only be assigned to one
-value.
-
For the function
is the name of the function, \(x\) is the domain value variable, and
is the range value variable.
-
The vertical line test is a test that can be done on the graph of a relation to determine if it is a function.
5.8
Graphing Functions
-
Every linear function can be graphically represented by a unique line that shows all the solutions of the equation.
-
The points where the graph of a line intersects the \(x\)-axis and
-axis are called the intercepts of the line.
-
Most lines will have one \(x\)-intercept and one
-intercept. Only if the line is straight vertical (no
-intercept) or straight horizontal (no \(x\)-intercept) will it not have both intercepts. Note that a line that is straight vertical is not a function, but a line that is straight horizontal is a function.
-
Since any two points determine a straight line, any linear function can be graphed if both intercepts are known.
-
The slope of a linear function is the ratio of the vertical change divided by the horizontal change. It is often referred to as
.
-
A formula for finding the slope of linear functions is
for any two points of the linear function
and
.
5.9
Systems of Linear Equations in Two Variables
-
To solve a system of linear equations means finding the point or points where the two linear equations intersect.
-
Two lines can intersect at one point, no points if they are parallel, or every point if they are the same equation.
-
Systems of linear equations can be solved by graphing, by using substitution, or by using the elimination method.
5.10
Systems of Linear Inequalities in Two Variables
-
To solve a system of linear inequalities means to find the area(s) where the points in that area make all the linear inequalities true.
-
Systems of linear inequalities can be solved by graphing the linear equations associated with the inequalities, then 'testing' points to see whether the values of the point make the equation true or not.
5.11
Linear Programming
-
Linear programming is a mathematical technique to solve problems involving finding maximums or minimums where a linear function is limited by various constraints.
-
An objective function is a linear function in two or more variables that describes the quantity that needs to be maximized or minimized.
-
In linear programming, a constraint is a restriction that affects the maximum or minimum values of an objective function.
-
Through the creation of objective functions and restraints, a linear system can be developed and solved through linear programming.


23.2: Intellectual Property
Intellectual Property is a form of intangible property representing the commercially valuable product of the human mind. IP can be in either an abstract or concrete form. For example, a composer may have IP interests to both the abstract sound of the music he or she composed, as well as the concrete sheet music that instructs musicians how to play the musical composition.
Protection of IP rights generally fall into one of four categories: patents, trade secrets, copyrights, and trademarks.
| Patent | Trade Secret | Copyright | Trademark | |
| Protects | Inventions & methods | Valuable secrets that give a business a competitive advantage | Tangible expression of an idea; but not the idea itself | Words & symbols used to identify products or services |
| Elements | Novel & non-obvious | Distinctiveness | Original expression | Reasonable measures in place to protect secrecy |
| Filing Requirements | Application must be approved by USPTO | Information must be kept secret but no formal process is necessary | Protection is automatic once creative expression takes tangible form; does not have to be filed with the CO but receives greater protection if it is | Mark is used in commerce; does not have to be filed with USPTO but receives greater protection if it is |
| Duration | 20 years (14 years for a design patent) | Forever, as long as information is kept secret | 70 years after death of author, 95 years from publication, or 120 years from creation (whichever is shorter) | 10 years but can be renewed an unlimited number of times as long as mark is still being used |
| Type of Law | Federal only | State (Economic Espionage Act may provide some federal protection depending on facts of case) | Federal only | Federal and state |
| Type of Enforcement Action | Infringement | Misappropriation | Infringement | Infringement & dilution |
| Criminal Liability | No | Yes | Yes | Yes |
| Expensive | Yes | No | No | Maybe (depends on policing costs) |
| Search Required for Existing IP Holders | Yes | No | No | Yes |
Different types of IP may attach to aspects of the same product or service. For example, Coca-Cola has trademarks for its name and logo, a patent for the shape of its original glass bottle, a copyright for its commercial jingle, and a trade secret for its cola recipe.
Figure 23.1 Examples of Coca-Cola trademarks


4.7: Other States of Consciousness
Learning Objectives
- Define hypnosis and meditation
- Understand the similarities and differences of hypnosis and meditation
Our states of consciousness change as we move from wakefulness to sleep. We also alter our consciousness through the use of various psychoactive drugs. This final section will consider hypnotic and meditative states as additional examples of altered states of consciousness experienced by some individuals.
Hypnosis
Hypnosis is a state of extreme self-focus and attention in which minimal attention is given to external stimuli. In the therapeutic setting, a clinician may use relaxation and suggestion in an attempt to alter the thoughts and perceptions of a patient. Hypnosis has also been used to draw out information believed to be buried deeply in someone’s memory. For individuals who are especially open to the power of suggestion, hypnosis can prove to be a very effective technique, and brain imaging studies have demonstrated that hypnotic states are associated with global changes in brain functioning (Del Casale et al., 2012; Guldenmund, Vanhaudenhuyse, Boly, Laureys, & Soddu, 2012).
Historically, hypnosis has been viewed with some suspicion because of its portrayal in popular media and entertainment. Therefore, it is important to make a distinction between hypnosis as an empirically based therapeutic approach versus as a form of entertainment. Contrary to popular belief, individuals undergoing hypnosis usually have clear memories of the hypnotic experience and are in control of their own behaviors. While hypnosis may be useful in enhancing memory or a skill, such enhancements are very modest in nature (Raz, 2011).
How exactly does a hypnotist bring a participant to a state of hypnosis? While there are variations, there are four parts that appear consistent in bringing people into the state of suggestibility associated with hypnosis (National Research Council, 1994). These components include:
- The participant is guided to focus on one thing, such as the hypnotist’s words or a ticking watch.
- The participant is made comfortable and is directed to be relaxed and sleepy.
- The participant is told to be open to the process of hypnosis, trust the hypnotist and let go.
- The participant is encouraged to use his or her imagination.
These steps are conducive to being open to the heightened suggestibility of hypnosis.
People vary in terms of their ability to be hypnotized, but a review of available research suggests that most people are at least moderately hypnotizable (Kihlstrom, 2013). Hypnosis in conjunction with other techniques is used for a variety of therapeutic purposes and has shown to be at least somewhat effective for pain management, treatment of depression and anxiety, smoking cessation, and weight loss (Alladin, 2012; Elkins, Johnson, & Fisher, 2012; Golden, 2012; Montgomery, Schnur, & Kravits, 2012).
Some scientists are working to determine whether the power of suggestion can affect cognitive processes such as learning, with a view to using hypnosis in educational settings (Wark, 2011). Furthermore, there is some evidence that hypnosis can alter processes that were once thought to be automatic and outside the purview of voluntary control, such as reading (Lifshitz, Aubert Bonn, Fischer, Kashem, & Raz, 2013; Raz, Shapiro, Fan, & Posner, 2002). However, it should be noted that others have suggested that the automaticity of these processes remains intact (Augustinova & Ferrand, 2012).
How does hypnosis work? Two theories attempt to answer this question: One theory views hypnosis as dissociation and the other theory views it as the performance of a social role. According to the dissociation view, hypnosis is effectively a dissociated state of consciousness, much like our earlier example where you may drive to work, but you are only minimally aware of the process of driving because your attention is focused elsewhere. This theory is supported by Ernest Hilgard’s research into hypnosis and pain. In Hilgard’s experiments, he induced participants into a state of hypnosis, and placed their arms into ice water. Participants were told they would not feel pain, but they could press a button if they did; while they reported not feeling pain, they did, in fact, press the button, suggesting a dissociation of consciousness while in the hypnotic state (Hilgard & Hilgard, 1994).
Taking a different approach to explain hypnosis, the social-cognitive theory of hypnosis sees people in hypnotic states as performing the social role of a hypnotized person. As you will learn when you study social roles, people’s behavior can be shaped by their expectations of how they should act in a given situation. Some view a hypnotized person’s behavior not as an altered or dissociated state of consciousness, but as their fulfillment of the social expectations for that role.
Meditation
Meditation is the act of focusing on a single target (such as the breath or a repeated sound) to increase awareness of the moment. While hypnosis is generally achieved through the interaction of a therapist and the person being treated, an individual can perform meditation alone. Often, however, people wishing to learn to meditate receive some training in techniques to achieve a meditative state. A meditative state, as shown by EEG recordings of newly-practicing meditators, is not an altered state of consciousness per se; however, patterns of brain waves exhibited by expert meditators may represent a unique state of consciousness (Fell, Axmacher, & Haupt, 2010).
Although there are a number of different techniques in use, the central feature of all meditation is clearing the mind in order to achieve a state of relaxed awareness and focus (Chen et al., 2013; Lang et al., 2012). Mindfulness meditation has recently become popular. In the variation of meditation, the meditator’s attention is focused on some internal process or an external object (Zeidan, Grant, Brown, McHaffie, & Coghill, 2012).
Meditative techniques have their roots in religious practices, but their use has grown in popularity among practitioners of alternative medicine. Research indicates that meditation may help reduce blood pressure, and the American Heart Association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension, although there is not sufficient data for a recommendation to be made (Brook et al., 2013). Like hypnosis, meditation also shows promise in stress management, sleep quality (Caldwell, Harrison, Adams, Quin, & Greeson, 2010), treatment of mood and anxiety disorders (Chen et al., 2013; Freeman et al., 2010; Vøllestad, Nielsen, & Nielsen, 2012), and pain management (Reiner, Tibi, & Lipsitz, 2013).
Summary
Hypnosis is a focus on the self that involves suggested changes of behavior and experience. Meditation involves relaxed, yet focused, awareness. Both hypnotic and meditative states may involve altered states of consciousness that have potential application for the treatment of a variety of physical and psychological disorders.
Glossary
- hypnosis
- state of extreme self-focus and attention in which minimal attention is given to external stimuli
- meditation
- clearing the mind in order to achieve a state of relaxed awareness and focus
Contributors and Attributions
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6.1: Love and Attraction Theory
One of the most important factors in interpersonal attraction is similarity : the more similar two people are in attitudes, background, and other traits, the more probable it is that they will like each other. Contrary to popular belief, opposites do not usually attract. Although physical attraction may take precedence in the early stages of a romantic relationship, similarity and other compatibility factors become more important later on.
Sternberg’s Triangular Theory of Love
Psychologist Robert Sternberg describes love based on combinations of three components: passion, intimacy, and commitment. When two people share all three, they are said to be in a state of consummate love. This combination is considered to be an ideal type of love, but is also relatively difficult to maintain for a long period of time.
Intimate love encompasses the close bonds of loving relationships. Intimate love felt between two people means that they each feel a sense of high regard for each other. They wish to make each other happy, share with each other, be in communication with each other, help when one is in need. A couple with intimate love deeply values each other. Intimate love has been called the “warm” love because of the way it brings two people close together.
Passionate love is based on drive. Couples in passionate love feel physically attracted to each other. Sexual desire is typically a component of passionate love. Passionate love is considered the “hot” component of love because of the strong presence of arousal between two people.
Commitment, or committed love, is for couples who are committed to being together for a long period of time. Something to note about commitment, however, is that one can be committed to someone without feeling love for him or her, and one can feel love for someone without being committed to him or her.
- Attraction and Love Theory. Authored by : Boundless. Located at : https://www.boundless.com/psychology/textbooks/boundless-psychology-textbook/social-psychology-20/social-behavior-105/attraction-loving-397-12932/ . License : CC BY-SA: Attribution-ShareAlike


6.3: Case Study 3- Pregnant Adult Client (continued)
1. What needs to be considered when interpreting the vital signs findings of this pregnant client?
- Even though the client’s reason for seeking care is shortness of breath, the Sp02 saturation is within the normal range and the respiratory rate is in the normal range. In the third trimester, it is common for the fetus to push against the diaphragm, causing it to raise and put pressure on the lungs resulting in the pregnant woman feeling short of breath.
- When measuring respiration, the healthcare provider assesses the depth of breathing: breathing is usually shallow in later-term pregnancy.
- Always take a fetal heart rate when taking a pregnant woman’s vital signs.
2. Which vital sign findings are considered abnormal for this pregnant client? What medical terminology is used to define/label these findings?
The blood pressure is elevated because the systolic pressure is 134 and the diastolic pressure is 90. Healthcare providers should first repeat the blood pressure reading for accuracy. They should also review the pregnant woman’s blood pressure levels pre-pregnancy and during previous pregnancies to determine the significance of these values. These considerations are important because slightly high systolic and diastolic blood pressure levels in later-term pregnant women can require an urgent referral for an obstetrical assessment.
The blood pressure is repeated on both arms after the client is at rest for five minutes to confirm accuracy.
1435 hrs, August 14, 2018
- BP 132/90 mm Hg right arm in sitting position, P 92 (radial)
- BP 128/88 mm Hg left arm in sitting position, P 88 (radial)
Try to answer the following questions.
- Which blood pressure do you record?
- Is the variation between the arms of concern?