A Simple Guide to Writing an Essay on Gone Girl

A Simple Guide to Writing an Essay on Gone GirlGone Girl is a novel by Gillian Flynn that takes place in New York. Many essay topics can be used to capture the essence of the book. Some of the topics include the study of crime, mental health and media. These topics will not only be used as background information, but also as supplemental sources to help you write a Gone Girl essay.Gone Girl essay topics include media coverage of crime and violence, especially murder. This will provide the basis for an excellent essay on media coverage of crime reporting itself. If you want to delve into the psychology of the crime itself, look at media coverage of divorce, custody battles and the psychological factors that drive the news.There are many different kinds of crimes. How much more so if the crime takes place in the city of New York? This will create interesting essay topics. For example, a student may look at what happens to people who commit acts of violence in the city of New York in th e way of gangs. Gangs come in many shapes and sizes and it's worth looking at this as a case study in psychopathy.There are other essay topics that go well with Gone Girl. A good example is a look at the role that media play in the lives of the rich and famous. Sometimes crime reporting may be more entertaining than the crime itself and this is a good topic to discuss in a Pulitzer prize-winning essay.You may want to look at the impact of the media coverage of the divorce of the Lady Gaga and John Mayer. The press greatly covered the marriage and divorce, but they failed to cover any of the personal information that was made public. Why? This led to all the world knowing that the man who left his wife in a cruise ship had been involved in multiple relationships, including the marriage of the pop star.The ultimate example of how media coverage of divorce affects society is seen in the plight of Kim Davis, who has refused to issue marriage licenses to same sex couples. A psychologist looked at how the press had reported the case and concluded that it was important because it highlighted the dangers of intolerance. The student could consider this topic by looking at the extreme reaction to the case in the news.Another essay topic that can apply to Gone Girl is the issue of mental health in this case. A teenager in a young woman who has serious mental health issues, has many lessons to teach us.Students are often challenged by writing, in that they have to choose a lesson or subject and learn to interpret the lesson that they are reading. One lesson would be that men can act abusively, while another might be about the detrimental effects of stereotypes on people's lives. There are many things that can be learned from essays, so always take your time when doing these and make sure that they are worthwhile.

Things to Expect From the USC Video Essay Samples

Things to Expect From the USC Video Essay SamplesIf you're wondering what to expect from the USC video essay samples, they are a great way to learn more about different topics and what the different universities are all about. If you're interested in taking this online test, you will want to know how to go about it, so here are some tips for you.First, you have to make sure that you are studying the right courses at USC. There are a few of them and they all teach different subjects, which can be intimidating when you first start taking them.Of course, before you even go to USC, you'll need to make sure that you're already taking the right classes. USC offers many different kinds of classes, so you might want to find a class that's very popular and that will help you understand the subject of your choice. Once you've picked a class, you can begin looking into the USC video essay samples.You should also make sure that you're studying each of the classes at USC, but in different ways. S ometimes you might take a full course at USC and other times you might only take one or two classes there. Take the time to read about the USC classes, though, and see what the different ones are all about.Once you've taken these classes at USC, you should find that you are ready to go on with the USC video essay samples. Make sure that you have a background in the material that you're going to study, because there will be lots of questions that you will need to answer.This is where you will find out about different topics like writing essays, history, math, social studies, or business. It might seem easy, but when you're first starting to do this, you'll need to learn how to do all of the different subjects in a logical and organized way.Once you get used to the coursework at USC, you'll want to continue your work by putting it to a test. This is where you will get to make your score known, so make sure that you're ready to take the test.

Essay on The Dust Bowl - 762 Words

GKE Task 1 A. Significant environmental /geographical factors that contributed to the development or expansion of the United States: 1. The Dust Bowl Farmers began to plow and plant wheat crops. When World War 1 began the massive wheat crops helped feed many Americans that in another part of the country try where in the beginning of a depression that was caused by the war. The wheat crops also helped feed numerous nations overseas. A drought that began in the beginning of the 1930’s persisted and was leading things in a very negative direction. No matter the circumstances farmers kept on farming hoping the rain would soon come back. In 1931 the rain just stopped and the farmers still kept trying to farm. Without adequate†¦show more content†¦B. One significant environmental or geographical factor that contributed to the development or diffusion on one human society from: Egypt’s geographic characteristics that affect the development of human society were the Nile River. The Nile River made the ability of a large population in a desert region. Also, the Nile would flood bringing rich soil to the lower regions of the rivers course. The crops in this area did well and the flooding helped set a steady schedule for planting and harvesting and because of this the societies remained stable and had extra time to develop art and science in their culture. The people of Egypt also had access to some of the largest quantities of stone such as granite. This began the development and building of pyramids instead of mud buildings. The Nile River played a huge role in the development of Egypt. 1. Cultural diffusion between Mesopotamia and Egypt. The social and politically elite from Egypt sought exotic goods and items of symbolic prestigiousness from India (Mesopotamia). For example: Mesopotamian cylinder seals and Afghan Lapis Lazuli which were found in early Egyptian tombs. Not only was there trades of items but there was trades of ideas. Egyptians imitated the Lapis Lazuli in their Egyptian Faience (ceramic of glass). Due to the Lapis Lazuli being very rare and expensive the Egyptian felt they could benefit from doing this. The Faience represents the first knownShow MoreRelatedThe Dust Bowl1192 Words   |  5 Pagesat a full moon. When he reached his house, his father rushed him inside. The first of many dust storms hit and the period known as the Dust Bowl began. The Dust Bowl was a brutal time period in Midwestern history; farmers were pushed off their land and forced to find new homes in new states. On a website called Drought Disasters, sponsored by Browing University, it was written â€Å"the seeds of the Dust Bowl may have been sown during the early 1920s. However, overproduction of wheat coupled with theRead MoreThe Dust Bowl 950 Words   |  4 PagesThe Dust Bowl was a treacherous storm, which occurred in the years of the 1930’s, which affected the Midwestern people, an example the farmers, which taught us new technologies and methods of farming. John Steinbeck wrote in his novel from 1939 The Grapes of Wrath: And then the dispossessed were drawn west- from Kansas, Oklahoma, New Mexico, Texas; from Nevada and Arkansas, families, tribes, dusted out, Caravans, carloads, and homeless. Totals of 20,000, 50,000, 100,000, and 200,000 people. TheyRead MoreDust Bowl Essay1233 Words   |  5 PagesThe Dust Bowl was a treacherous storm, which occurred in the 1930s, that affected the midwestern people, for example the farmers, and which taught us new technologies and methods of farming. As John Steinbeck wrote in his 1939 novel The Grapes of Wrath: And then the dispossessed were drawn west- from Kansas, Oklahoma, Texas, New Mexico; from Nevada and Arkansas, families, tribes, dusted out. Carloads, caravans, homeless and hungry; twenty thousand and fifty thousand and a hundred thousand and twoRead MoreThe Dust Bowl Essay1436 Words   |  6 PagesOklahoma, Kansas, Colorado, and New Mexico—the Dust Bowl was a time where over 100 million acres of topsoil were stripped from fertile fields leaving nothing but barren lands and piles of dust everywhere (Ganzel). While things were done to alleviate the problem, one must question whether or not anyone has learned from this disaster. If not, one must look into the possibility that the United States may be struck by such a destructive drought as the Dust Bowl, if not a worse one that would leave us withRead MoreThe Dust Bowl Essay1038 Words   |  5 PagesThe Dust Bowl The Dust Bowl was the darkest moment in the twentieth-century life of the southern plains, (pg. 4) as described by Donald Worster in his book The Dust Bowl. It was a time of drought, famine, and poverty that existed in the 1930s. Its cause, as Worster presents in a very thorough manner, was a chain of events that was perpetuated by the basic capitalistic societys need for expansion and consumption. Considered by some as one of the worst ecological catastrophes in theRead MoreThe Cause Of The Dust Bowl1181 Words   |  5 Pages16 October 2017  Outline Thesis: The Dust Bowl of the 1930s forever changed how Americans thought of and treated our farm lands in the Great Plains. Introduction I. Causes of the Dust Bowl A. The Drought in the Great Plains B. Improper Tending of the Land II. People Affected by the Dust Bowl A. Lawrence Srobin, Aris D. Carlson, and John Steinbeck B. Statistics of the Damage C. Farmer’s Problems Before the Dust Bowl III. FDR’s Fix for the Dust Bowl A. Strategic Planting of Trees in theRead More Dust Bowl Essay915 Words   |  4 Pagesanswers.com, a dust bowl is a region reduced to aridity by drought and dust storms. The best-known dust bowl is doubtless the one that hit the United States between 1933 and 1939. One major cause of that Dust Bowl was severe droughts during the 1930’s. The other cause was capitalism. Over-farming and grazing in order to achieve high profits killed of much of the plain’s grassland and when winds approached, nothing was there to hold the devastated soil on the ground. The Dust Bowl affected the GreatRead MoreThe Great Depression And Dust Bowl1165 Words   |  5 PagesDepression/Dust Bowl The ‘Dirty Thirties’ is perhaps one of the most known time periods in American History. During the 1930s, the worst and longest drought occurred in the United States, this was also know as the Dust Bowl. According to Christopher Klein, the Dust Bowl is considered both a man-made and natural disaster. In fact, many events contributed to the Dust Bowl such as poor farming techniques, a severe drought, and economic depression. One of the main causes of the Dust Bowl was the poorRead MoreThe Dust Bowl and Agriculture Essay1070 Words   |  5 Pages One has not experienced the life of living in dirt until he has been in the dust bowl. It was a decade-long dust storm that impacted hundreds of farmers and their farmlands. Hardship was among one of the influences of the storm, which affected both farm workers and city folks. The storm also brought the elements of destruction and darkness, which reigned chaos across the Plains. Together, these issues gave the storm its popular name, â€Å"black blizzard† (Documentary, 2014). Such a name was given dueRead MoreThe Black Blizzard And The Dust Bowl1570 Words   |  7 PagesDuring the Dust Bowl many people and kids have suffered, many lost their home and their towns got ruined. One of the people who has suffered in the Dust Bowl is Ashton. When Ashton went to his school he was immediately pulled in by his teacher Mrs. Kam. He was then told that the entire middle east was affected by the Dust Bowl and that a black blizzard will hit very soon. Then the winds outside started to get faster, the windows getting hit by all the dust gathered from the storm, but luckily for

North Korea Free Essays

From an early date Korean political culture is characterised by isolationism and a strong desire to maintain the country’s independence (â€Å"Kim Il Sung†, para 2). Relationship with its neighbouring countries are poor and with the countries in the West, are almost equal to zero. Even with such relationships, Korea has been unable to stop the encroachment   of its neighbours. We will write a custom essay sample on North Korea or any similar topic only for you Order Now It was made the Japenese protectorate in 1905 and later the number of   Japenese immigrants in Korea have risen so much that the Koreans have become second-class citizens within their own land. Kim Il Sung Kim Il Sung’s real name was Kim Song Ju. He was born in Pyongyang on 15th April, 1912. At the age of   17 he was jailed for being a part of   a student political group led by the South Manchurian Communist Youth Association. After his release from jail he joined the Anti-Japanese United Army. He emerged as a significant leader in this time period and later changed his name to Kim Il Song, in honor of his uncle who participated in nation-wide protests against the Japenese in 1919. Kim Il Sung as a leader After the Second World War the Korean peninsula was divided into two parts, North and South. North Korea was sponsored by Russia while the South was aided by USA. Kim Il Sung was selected to take charge of the formation of a provisional government for the North. Under his leadership the Korean Workers Party was inaugurated. A number of reforms were introduced to the North, including an eight-hour working day, equality of the sexes, and suppression of religion. Land and wealth formerly belonging to the Japanese or to enemies of the regime was confiscated and redistributed, industry was nationalised, and Soviet-style economic planning was initiated (â€Å"Kim Il Sung†, para 12) His opponents within the party were purged to secure his absolute rule. Kim led the military committee, coordinating action against the South Koreans in 1950. With carefully prepared plans and without any warning to the South Koreans, his army swarmed into the South. The war continued for three years and during these three years about three million people lost their lives. Between 600,000 and one million North Koreans needlessly starved to death due to the economic legacy of Kim’s regime. Kim claimed to have won that war but no peace treaty was ever signed. Kim Il Sung, through land reclamation, gave priority to increased agricultural production. He emphasised on trade, developed the country’s infrastructure, and encouraged people to rely on domestically produced equipment. He discouraged them to even think about foreign aid and taught them that self reliance should be the key goal to everyone’s success. From 1972 onwards, North Korea became poorer because it could not afford to buy advance technology from the West and its industrial production declined. A personality cult had glorified Kim, but by the mid-1990s the rapid economic growth of North Korea’s early years had given way first to stagnation and then to hardship, and there was widespread dissatisfaction with the repressive regime (Scalapino Lee p.175) When he died in 1994, the country lost its venerated founding leader. Just a few years earlier, its powerful alliances had evaporated with the fall of the Soviet bloc and China’s move toward a market-based system. The economy was on the rocks and energy and food were in short supply. A series of weather disasters, combined with an inefficient state-run agricultural system, further eroded the food supply, leading to mass starvation (â€Å"Dear Leader or Demon†, para 12) Kim Jong Il Kim Il Sung’s successor, Kim Jong Il, was born on 16th June, 1942. Growing up in a time when anti Japanese revolutionary struggle was at its peak, Kim Jong Il cultivated uncommon characteristics and qualities through his experiences of his real life and practical activities. During his training period under his father’s tutelage in the 1970s, he was often referred to as the â€Å"Party Center,† and he launched a number of campaigns to take over the daily operations of the Party. When he came to power on the death of his father in 1997, Kim Jong-Il ruthlessly set about establishing his own authority. His government is said to be extremely secretive and brutal to dissidents. Kim’s disastrous agricultural and economic policies have caused his people to suffer under one of the world’s longest, deadliest famines. On the domestic front, Kim has given occasional signs that he favors economic reforms similar to those carried out in China by Deng Xiaoping. But at home he has done little or nothing to relax the absolute control of the state and party over all aspects of economic life. He has certainly given no sign of considering the de- collectivization of agriculture, which was the foundation of Deng’s reforms (Chong-Il p. 84). With the sort of image Kim Jong Il carries around with him many analysts thought that the communist regime would finally come to an end in North Korea, but it is almost a decade since he has been in power and still ruling over North Korea and extending his father’s philosophy of Juche. Kim Il Sung’s style of ruling his country according to Stalinism was submerged into his Juche philosophy and later it subordinated into a more militant theme of Kim Jong Il’s Red Banner Policy. Kim Jong Il has been accused of being involved in two bombings; one in Rangoon in 1983 and another in 1987, which killed all passengers in a South Korean airplane.   No evidence directly links Kim Jong Il to the bombings, however, and some analysts believe his father was still firmly in control of international activities throughout the 1980s, while giving his son more power over domestic affairs. He spends more than 25% of his country’s annual GNP on the military while most of the citizens go hungry. Jong Il has also proved himself to be a movie lover and has directed a couple of movies himself and written six operas, while his scientists make nuclear war heads. Kim Il Sung as a Communist In the words of Morelly, the author of the influential treatise Le Code de la Nature, published in 1755: †The only vice which I know in the universe is avarice; all the others, whatever name one gives them, are merely forms, degrees of it. . . Analyze vanity, conceit, pride, ambition, deceitfulness, hypocrisy, villainy; break down the majority of our sophisticated virtues themselves, [they] all dissolve in this subtle and pernicious element, the desire to possess.† Such economically determined psychology lies at the root of every socialist and communist doctrine (Pipes 2001). Kim Il Sung was one man who was obsessed with gaining power over not only his own people but he went to extremes and even created his own religion so that the people should start worshipping him. When workers of the Korean Workers Party tried to over throw Kim in 1953, the eleven conspirators, who failed in their coup, were not only executed but their properties were confiscated as well. He purged his opponents in all possible ways in order to ensure that he stays in power. Kim Il Sung’s personality was a true speciman of a communist. Russia had an influence over the political system of North Korea right from the onset. Kim Il Sung was highly impressed by the communist rule in Russia and he developed a Marxist-Leninist political ideology that emphasises the need for autonomy and patriotic self-reliance. Called ‘Juche’, or ‘Kim Il Sung Thought’, the ideology demanded total loyalty to the paramount leader and the â€Å"religion of Kim Il Sungism†, and stressed on the benefits of sacrifice, austerity, discipline, dedication, unity, and patriotism. It has been   described as â€Å"encyclopedic thought which provides a complete answer to any question that arises in the struggle for national liberation and class emancipation, in the building of socialism and communism.† The practical effect of Juche was to seal the DPRK off from virtually all foreign trade (â€Å"Kim Il Sung†, para 38). Kim Il Sung’s personality cult was similar to that of   Stalin. In that he was accused of tampering history, tyranny and injustice and self glorification. Kim’s zeal for communsim is evedient from the fact that the North Korean media, which is owned and controlled by the state, promoted Kim’s image as an infallible genius and the driving force behind the resistance to the Japanese and the liberation of the North. Kim had more than 34,000 monuments of himself erected throughout the country. Practice of any other religion except the worship of Kim Il Sung   was outlawed. The official calender started from his birthday and that day was an official holiday too. Kim Il Sung was so obssessed with his ideals that he started to believe in himself in a godly manner. The painters and artists of the country were instucted to paint only his personality in a pompous manner and project it to the rest of the country. All this was believed, by   Kim Il Sung, as not a forceful extension of his one-man show personality to the masses but he had actually started to live in a pseudo-shell of belief   that the citizens of his country actually adored him.   In my point of view the personality of Kim Il Sung was a fragment and extension of Stalin. This is supported by the fact that he not only tried to follow him in the way he ruled North Korea but he also sought his help and looked up to him in various matters of   state. In the Korean War of   1950-1953, while South Korea was supported by USA, North Korea was aided by USSR primarily because it was a communist country. Kim Jong Il has proved himself to be a true replica of his father. He has not only ensured the continuity of his father’s policies and reforms but he has also gone to the extent of further subjugating the agrigarian community by spending more on his military might. He has promoted his father’s personality cult by making it a compulsion for each citizen to visit Kim Il Sung’s moseleum atleast once each year and also by paying tribute to the Senior Kim by wearing lapel badges and all those who dare to offend are sent to concentration camps within the country. Kim Il Sung has become so much part of   a North Korean’s daily life that to reject Kim is more like rejecting his own self. Conclusion With 1% of the citizens in concentration camps and 25% of GNP being utilized to further the ruler’s military prowess, North Korea is definitely not in pursuit of a prosperous future. While the exiguity of the basic facilties of life and the over all plight of the nation is being blamed on the West in general and the US in particular, nobody dares to question the extravagent life of the dictators of North Korea. North Korea is going through its worst era right now; with its predominent support/donor, in the shape of USSR, disintegrated and with the West pressing hard on it on the contentious issue of its nuclear programme. References Clark, Carol. (2001). Kim Jong Il, â€Å"Dear Leader or Demon?† Retrieved 13 May 2006 from http://www.cnn.com/korea/CNN–InDepth Korea at 50 – Leader Profiles Kim Jong II.htm Chong-Il, Kim. (1986). Life and literature. Pyongyang, Korea: Foreign Languages Publishing House. Harris, Bruce. (2004). â€Å"Kim Il Sung†. Retrieved 12 May 2006 from http://www.moreorless.au.com Pipes, Richard. (2001). Communism: A History. Random House Publishing Group. R. A. Scalapino and C. S. Lee. (1992). Communism in Korea. Seoul: Ilchokak. How to cite North Korea, Essay examples

Unnatural Causes free essay sample

Whereas, Tondra Young’s district, district 24, has an average combined household income of $70,000/annually and 15% of the population has a college degree. The life expectancy of district 24 is 75 years, that’s 7 years less than district 16. In district 21, Corey Anderson’s district, the average combined household income is less than $50,000/ annually and only 5% of the districts population have college degrees. Consequently, the life expectancy of district 21 is 70 years, 5 years less than district 24 and 12 years less than district 16 (Adelman 2008).The results of this study are indisputable: there is an obvious correlation between social/economic status and health status. As each districts average income and education level decrease, average life expectancy coincides. This leads to the next question: why does social and economic status so greatly influence health status? One could supply many obvious reasons for this correlation in health care; this correlation, as illustrated above, shows research-based evidence that being of lower economic and social standing places individuals at an increased risk for health complications. We will write a custom essay sample on Unnatural Causes or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page The more obvious explanations for this occurrence could be that individuals with decreased economic and social status would have less access to health care related to diminished finances, or they would have unhealthy diets related to higher costs of healthy foods, or due to their lack of substantial education they would, in turn, lack in knowledge of health care practices. Oddly enough, these factors are not the major contributor. The reason for the correlation between health care and social/ economic status, which the documentary primarily focuses on, can be narrowed down to one culprit: stress.According to the documentary, it is the increased stress level in lower income population that accounts for the increase in morbidities. The result of chronic stress on the body is a chronic release of the body’s stress hormone: cortisol (Adelman 2008). When the body accumulates too much cortisol it leads to a decrease in immune system function and thus, creating an increase in sickness. Regardless of one’s social status or stress level, there is one more disturbing factor that prevents a person from achieving optimal health care: race.Racial discrimination has been linked with high blood pressure, increased rates of coronary artery disease and infant death. African Americans, for example, have higher rates of many chronic diseases and die earlier that Caucasians, regardless of the social gradient. In a National study it was found that over 83,000 excess deaths occurred per year in the African American community alone! â€Å"That’s the equivalent of a major airline filled with Black passengers falling out of the sky every single day, every year. † (Adelman 2008). There are many ways we can improve health care equity.The two main things we can do are to identify health determinants and create public health programs. Public health programs traditionally address population health through measures to prevent and control health problems; it is also important that they identify the structural and intermediate determinants that lead to ill health (CDC 2012). We as nurses can help with this approach as we are always observing and assessing. It is our jobs as nurses to identify social determinants in our patients and identify ways we can help them individually. For example, a patient has been prescribed ten different medications and the patient tells you she can only afford a few of them and will only take the important ones. It is our jobs to help this woman correctly identify which of the medications is indeed the most important. Nurses can always promote community health with programs such as: blood pressure screenings, maternity lectures etc. Making simple and affordable preventative care measure available to the community is the most important thing we can do to improve health!

Until When essays

Until When essays The 2000 Presidential election has brought much attention to itself. While a slew of lawyers try to cheat their respective political figurehead into the White House, the topics discussed during the debates have been put aside. Affirmative action and education touch upon a delicate subject, which hinders the fundamental progress of our nation. Racism is nothing new to this country. In fact, much of the early development in America is the result of slavery. In this paper I will attempt to use the African American people as an example of how the deep roots of racism, like anything else, has evolved over time and has all but escaped America. Everyone is aware of the problems slavery has caused in the United States. It caused the nation to divide, as was the case in the Civil War. The wars conclusion granted the slaves freedom. Emancipation didnt necessarily end racism or better the African Americans situation. Now they had to fend for themselves with no economic support, education, or rights. As a whole, the development of the African American community had been suppressed and left a few steps behind. The Great Depression seemed to even out the economic side of things, making hardship more uniform for all. This led to the emergence of government assistance programs ( Mullings 1986: 41). The economic crisis that affected the United States during the Great Depression became the stage for the creation of many social programs geared towards providing economic relief to American families. The New Deal relief programs emerged in the 1930s. Racial discrimination insured that in many communities black families received smaller grants than white families. They were also excluded from many of the employment programs ( Bremner 1992: 11). Welfare at first seemed to be the opportunity to achieve financial security not dependence. After World War II, the economy improved and the majority of middle and upper class white Americ...

Maximum Likelihood Estimation Examples

Maximum Likelihood Estimation Examples Suppose that we have a random sample from a population of interest.  We may have a theoretical model for the way that the population is distributed.  However, there may be several population parameters of which we do not know the values.  Maximum likelihood estimation is one way to determine these unknown parameters.   The basic idea behind maximum likelihood estimation is that we determine the values of these unknown parameters.  We do this in such a way to maximize an associated joint probability density function or probability mass function.  We will see this in more detail in what follows.  Then we will calculate some examples of maximum likelihood estimation. Steps for Maximum Likelihood Estimation The above discussion can be summarized by the following steps: Start with a sample of independent random variables X1, X2, . . . Xn from a common distribution each with probability density function f(x;ÃŽ ¸1, . . .ÃŽ ¸k).  The thetas are unknown parameters.Since our sample is independent, the probability of obtaining the specific sample that we observe is found by multiplying our probabilities together.  This gives us a likelihood function L(ÃŽ ¸1, . . .ÃŽ ¸k)   f( x1 ;ÃŽ ¸1, . . .ÃŽ ¸k) f( x2 ;ÃŽ ¸1, . . .ÃŽ ¸k) . . .  f( xn ;ÃŽ ¸1, . . .ÃŽ ¸k) ÃŽ   f( xi ;ÃŽ ¸1, . . .ÃŽ ¸k).Next, we use Calculus to find the values of theta that maximize our likelihood function L.  More specifically, we differentiate the likelihood function L with respect to ÃŽ ¸ if there is a single parameter.  If there are multiple parameters we calculate partial derivatives of L with respect to each of the theta parameters.To continue the process of maximization, set the derivative of L (or partial derivatives) equal to zero and solve for theta.We can then use o ther techniques (such as a second derivative test) to verify that we have found a maximum for our likelihood function. Example Suppose we have a package of seeds, each of which has a constant probability p of success of germination.  We plant n of these and count the number of those that sprout.  Assume that each seed sprouts independently of the others.  How do we determine the maximum likelihood estimator of the parameter p? We begin by noting that each seed is modeled by a Bernoulli distribution with a success of p. We let X be either 0 or 1, and the probability mass function for a single seed is f( x ; p ) px (1 - p)1 - x.   Our sample consists of n  Ã‚  different Xi, each of with has a Bernoulli distribution.  The  seeds that sprout have Xi 1 and the seeds that fail to sprout have Xi 0.   The likelihood function is given by: L ( p ) ÃŽ   pxi (1 - p)1 - xi We see that it is possible to rewrite the likelihood function by using the laws of exponents.   L ( p )   pÃŽ £ xi (1 - p)n - ÃŽ £ xi Next we differentiate this function with respect to p.  We assume that the values for all of the Xi are known, and hence are constant.  To differentiate the likelihood function we need to use the product rule along with the power rule: L ( p )   ÃŽ £ xip-1 ÃŽ £ xi (1 - p)n - ÃŽ £ xi - (n - ÃŽ £ xi )pÃŽ £ xi (1 - p)n-1 - ÃŽ £ xi We rewrite some of the negative exponents and have: L ( p ) (1/p) ÃŽ £ xipÃŽ £ xi (1 - p)n - ÃŽ £ xi - 1/(1 - p) (n - ÃŽ £ xi )pÃŽ £ xi (1 - p)n - ÃŽ £ xi [(1/p) ÃŽ £ xi  - 1/(1 - p) (n - ÃŽ £ xi)]ipÃŽ £ xi (1 - p)n - ÃŽ £ xi Now, in order to continue the process of maximization, we set this derivative equal to zero and solve for p: 0 [(1/p) ÃŽ £ xi  - 1/(1 - p) (n - ÃŽ £ xi)]ipÃŽ £ xi (1 - p)n - ÃŽ £ xi Since p and (1- p) are nonzero we have that 0 (1/p) ÃŽ £ xi  - 1/(1 - p) (n - ÃŽ £ xi). Multiplying both sides of the equation by p(1- p) gives us: 0 (1 - p) ÃŽ £ xi  - p (n - ÃŽ £ xi). We expand the right hand side and see: 0   ÃŽ £ xi  - p ÃŽ £ xi  - p n pÃŽ £ xi   ÃŽ £ xi - p n. Thus ÃŽ £ xi p n and (1/n)ÃŽ £ xi   p.  This means that the maximum likelihood estimator of p is a sample mean.  More specifically this is the sample proportion of the seeds that germinated.  This is perfectly in line with what intuition would tell us.  In order to determine the proportion of seeds that will germinate, first consider a sample from the population of interest. Modifications to the Steps There are some modifications to the above list of steps.  For example, as we have seen above, is typically worthwhile to spend some time using some algebra to simplify the expression of the likelihood function.  The reason for this is to make the differentiation easier to carry out. Another change to the above list of steps is to consider natural logarithms. The maximum for the function L will occur at the same point as it will for the natural logarithm of L.  Thus maximizing ln L is equivalent to maximizing the function L. Many times, due to the presence of exponential functions in L, taking the natural logarithm of L will greatly simplify some of our work. Example We see how to use the natural logarithm by revisiting the example from above.  We begin with the likelihood function: L ( p )   pÃŽ £ xi (1 - p)n - ÃŽ £ xi . We then use our logarithm laws and see that: R( p ) ln L( p ) ÃŽ £ xi ln p (n - ÃŽ £ xi) ln(1 - p). We already see that the derivative is much easier to calculate: R( p ) (1/p)ÃŽ £ xi - 1/(1 - p)(n - ÃŽ £ xi) . Now, as before, we set this derivative equal to zero and multiply both sides by p (1 - p): 0 (1- p ) ÃŽ £ xi -  p(n - ÃŽ £ xi) . We solve for p and find the same result as before. The use of the natural logarithm of L(p) is helpful in another way.  It is much easier to calculate a second derivative of R(p) to verify that we truly do have a maximum at the point (1/n)ÃŽ £ xi   p. Example For another example, suppose that we have a random sample X1, X2, . . . Xn from a population that we are modelling with an exponential distribution.  The probability density function for one random variable is of the form f( x ) ÃŽ ¸-1 e -x/ÃŽ ¸ The likelihood function is given by the joint probability density function. This is a product of several of these density functions: L(ÃŽ ¸) ÃŽ   ÃŽ ¸-1 e -xi/ÃŽ ¸   ÃŽ ¸-n e -ÃŽ £ xi/ÃŽ ¸    Once again it is helpful to consider the natural logarithm of the likelihood function.  Differentiating this will require less work than differentiating the likelihood function: R(ÃŽ ¸) ln L(ÃŽ ¸) ln [ÃŽ ¸-n e -ÃŽ £ xi/ÃŽ ¸] We use our laws of logarithms and obtain: R(ÃŽ ¸) ln L(ÃŽ ¸) - n ln ÃŽ ¸Ã‚   -ÃŽ £xi/ÃŽ ¸ We differentiate with respect to ÃŽ ¸ and have: R(ÃŽ ¸)   - n / ÃŽ ¸Ã‚   ÃŽ £xi/ÃŽ ¸2 Set this derivative equal to zero and we see that: 0 - n / ÃŽ ¸Ã‚   ÃŽ £xi/ÃŽ ¸2. Multiply both sides by ÃŽ ¸2 and the result is: 0 - n ÃŽ ¸Ã‚   ÃŽ £xi. Now use algebra to solve for ÃŽ ¸: ÃŽ ¸ (1/n)ÃŽ £xi. We see from this that the sample mean is what maximizes the likelihood function.  The parameter ÃŽ ¸ to fit our model should simply be the mean of all of our observations. Connections There are other types of estimators.  One alternate type of estimation is called an unbiased estimator.  For this type, we must calculate the expected value of our statistic and determine if it matches a corresponding parameter.