Find the Inflection Points for the Normal Distribution

One thing that is great about mathematics is the way that seemingly unrelated areas of the subject come together in surprising ways. One instance of this is the application of an idea from calculus to the bell curve. A tool in calculus known as the derivative is used to answer the following question. Where are the inflection points on the graph of the probability density function for the normal distribution? Inflection Points Curves have a variety of features that can be classified and categorized. One item pertaining to curves that we can consider is whether the graph of a function is increasing or decreasing. Another feature pertains to something known as concavity. This can roughly be thought of as the direction that a portion of the curve faces. More formally concavity is the direction of curvature. A portion of a curve is said to be concave up if it is shaped like the letter U. A portion of a curve is concave down if it is shaped like the following ∠©. It is easy to remember what this looks like if we think about a cave opening either upward for concave up or downwards for concave down. An inflection point is where a curve changes concavity. In other words it is a point where a curve goes from concave up to concave down, or vice versa. Second Derivatives In calculus the derivative is a tool that is used in a variety of ways. While the most well-known use of the derivative is to determine the slope of a line tangent to a curve at a given point, there are other applications. One of these applications has to do with finding inflection points of the graph of a function. If the graph of y f( x ) has an inflection point at x a, then the second derivative of f evaluated at a is zero. We write this in mathematical notation as f’’( a ) 0. If the second derivative of a function is zero at a point, this does not automatically imply that we have found an inflection point. However, we can look for potential inflection points by seeing where the second derivative is zero. We will use this method to determine the location of the inflection points of the normal distribution. Inflection Points of the Bell Curve A random variable that is normally distributed with mean ÃŽ ¼ and standard deviation of ÏÆ' has a probability density function of f( x ) 1/ (ÏÆ' √(2 Ï€) )exp[-(x - ÃŽ ¼)2/(2ÏÆ'2)]. Here we use the notation exp[y] ey, where e is the mathematical constant approximated by 2.71828. The first derivative of this probability density function is found by knowing the derivative for ex and applying the chain rule. f’ (x ) -(x - ÃŽ ¼)/ (ÏÆ'3 √(2 Ï€) )exp[-(x -ÃŽ ¼) 2/(2ÏÆ'2)] -(x - ÃŽ ¼) f( x )/ÏÆ'2. We now calculate the second derivative of this probability density function. We use the product rule to see that: f’’( x ) - f( x )/ÏÆ'2 - (x - ÃŽ ¼) f’( x )/ÏÆ'2 Simplifying this expression we have f’’( x ) - f( x )/ÏÆ'2 (x - ÃŽ ¼)2 f( x )/(ÏÆ'4) Now set this expression equal to zero and solve for x. Since f( x ) is a nonzero function we may divide both sides of the equation by this function. 0 - 1/ÏÆ'2 (x - ÃŽ ¼)2 /ÏÆ'4 To eliminate the fractions we may multiply both sides by ÏÆ'4 0 - ÏÆ'2 (x - ÃŽ ¼)2 We are now nearly at our goal. To solve for x we see that ÏÆ'2 (x - ÃŽ ¼)2 By taking a square root of both sides (and remembering to take both the positive and negative values of the root  ±ÃÆ' x - ÃŽ ¼ From this it is easy to see that the inflection points occur where x ÃŽ ¼  ± ÏÆ'. In other words the inflection points are located one standard deviation above the mean and one standard deviation below the mean.

The Varying Intensity of the Cold War in Europe Essay

The Varying Intensity of the Cold War in Europe The German surrender on the 7th May 1945 marked the end of the Second World War in Europe and heralded the beginning of a new conflict. This conflict would develop into the Cold War between the two largest countries in the world at the end of the Second World War, the United States of America (USA) and The Union of Socialist Soviet Republics (USSR). This essay will examine the Cold War from a European perspective. It is important to examine the Cold War from a European Perspective because within Europe the USAs and the USSRs spheres of influence had a physical border. This physical border was christened the Iron Curtain by the former†¦show more content†¦The reasons behind the change in the USAs foreign policy from isolationist, which it was before the Second World War, to interventionist, which it became, were varied. They were led by a number of different fears. These included the size of the Soviet army the Red Army, which was the largest army on the planet, the fact that Europe was weak and therefore vulnerable to a communist take over and the development of technology particularly of Russian nuclear weapons. The Truman Doctrine led to an increase in intensity between the USA and the USSR as the USSR believed that the USA was attempting to spread capitalism across Europe particularly to the new Eastern Block countries. The USSR believed that governments within Europe should be communist and therefore allied with her against capitalism. The next occurrence, which led to an increase in tension between the USA and the USSR, was the announcement by the Americans of the Marshall Plan. The US Secretary of State George Marshall, in a speech made at Harvard on the 5th June 1947 announced the proposal, which later became the Marshall Plan. 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Behind the Beautiful Forever-.com

Question: Discuss about the Novel Behind the beautiful forever: Life, Death and Hope in a Mumbai City. Answer: Introduction The novel Behind the beautiful forever: Life, Death and Hope in a Mumbai City by Katherine Boo captures the life of various slum dwellers in the Mumbai city (Boo, 2014). She has tried to showcase the daily life and dreams of the people who live below the poverty line in the Indian city. It is a non-fictional book that portrays the real-life challenges of many people in the developing country. The thesis statement is the interconnected lives of various characters have been described in the story by Katherine. The story takes place in Annawadi. The story is unique since the author believes that every character represents a part of her. Living conditions and other aspects of life in the slum The non-fictional theme has been used by Katherine while writing the story so that the actual living conditions of the slum people can be portrayed in the most realistic manner. The daily issues and challenges that are faced by the residents such as poverty, illiteracy, hunger, dirt, disease, lack of hygiene, etc are highlighted in the book. Various characters have been introduced like Abdul, a person who sorts garbage in the specific slum, Karam and Zehrunisa, his parents, and Mirchi, his younger brother. Sunil is another young resident of Annawadi who looks for trash so that he can sell them to Abdul (Boo, 2014, pp 25). The different characters have been highlighted so that the living conditions of the people can be portrayed in an accurate manner. Different things like violence, crime, theft, malnutrition, etc that are faced by the inhabitants on a daily basis form the structure of the story. The story is an attempt to paint the life of people living in the slum setting. Key characteristic of the book which made the strongest impression The major characteristic of the book that makes the strongest impression on me as a reader is the integrity of Abdul (Boo, 2014, pp 53). Even though the young boy has gone through various challenges at a very young age due to the poverty setting, he believes in his hard work and dedication. He tries to support his entire family of eleven members by working as a hard working garbage sorter in the locality. He makes a genuine attempt to instill positive characteristics in another fellow slum dweller named Sunil. He advises Sunil and other boys to stop stealing. After he goes to the Dongri jail he himself stops purchasing any kind of recycled products that have been stolen. The main reason that makes it the key characteristic of the book is that integrity, dedication and hard work can be portrayed by any individual irrespective of their financial condition. In spite of having no financial stability, Abdul serves as the storys moral compass. His honest principles to lead life inspire the reader to act as a responsible individual. Describing the person Abdul is surely the key character in the story. He is the oldest son of the Husain family who knows his duties and responsibilities. He believes that by adopting an honest and hardworking approach he can go ahead in his life. He also tries to make positive changes around himself. He gives valuable advice to the fellow slum boys so that they would prohibit from stealing. His strong moral integrity shows that he is a dedicated individual who believes in following a principled path in spite of having various challenges in his life (Boo, 2014, pp ). Abdul has a big family which makes his financial condition even more challenging for him. Since he is a Muslim it adds on to his challenges in life. Various challenges in his life have been showcased in the book where it shows the ups and downs that he has gone through in his entire life. But the most influencing part is the dedication that is seen in the young boy at such an early age. He is seen taking good decisions which made him achieve a better place in his life. His sense of responsibility makes him unique and inspiring. Conclusion In the story, Katherine has very effectively captured the attention of the readers by showcasing the slum life of people in a realistic manner. The various living conditions and challenges that have been highlighted in the story are something that almost every individual can relate to at some point in time. The characters that have been featured in the book show genuine emotions and undergo real challenges. These aspects make the story very baffling and interesting. Reference Boo, K. (2014). Behind the beautiful forevers: Life, death, and hope in a Mumbai undercity. Random House Trade Paperbacks. pp 10-19

Medical School Personal Statement Examples - A Must Read

Medical School Personal Statement Examples - A Must ReadYour writing skills are absolutely worthless in giving you the guidelines to a stellar medical school personal statement. Let me explain.I know it's your writing. But that's how you get there. You write down what you think. After months of writing, you come to an ideal that just seems to be right.It isn't. And if you submit an application, it will get thrown out.Lets look at an example. Say your are in the process of applying to a med school. Your ad is advertised in a national school's advertising classifieds for students to apply to schools that will be accepted into medical school.You do not have the time or the patience to research letters of recommendation, because you are too busy with work, school, and you have family obligations. After weeks and months of writing, and still you find yourself in the awkward position of a little stammering, a little dyslexia, and trying to come up with just the right words to begin.Take a good hard look at your writing skills, and then seek out some examples of personal statements. It doesn't matter if it's a typical story, or a statement written in a story style. If you can come up with some example sentences, you will be able to prepare yourself much better for the interview. When the interviewer sees you, it will be easier for them to be able to articulate what you want to hear.To make your statement stand out, give one standard example of each type of statement. In the end, the interviewer will be looking for many different types of statements.Keep in mind, even if you've already chosen a medical school. There are many things that are critical for you to know. So, spend some time researching for examples of statements, and practice and improve your writing skills.

Funny Argumentative Essay Topics For College Students

Funny Argumentative Essay Topics For College StudentsSo you want to write a good argumentative essay topic for college students? Well, you have many more options than you think. Even though some of these topics will be unconventional, some of them will be quite conventional. The first thing to understand is that writing an argumentative essay topic for college students requires that you plan ahead of time.In the end, you are going to write the essay for someone who already has their head screwed on. The only way to make this easier is to focus on their point of view. Be sure to acknowledge their points of view as well as your own.But what if you don't like the point perspective? What if you feel the need to bring in something different? Don't fret because that is the great thing about writing essays. If you come up with something good, you can recycle it and make something even better.Now is the time to pick your topic. I suggest that you pick something completely out of left field. This way, you are not just sticking with one or two topic ideas but instead, can include a broad range of topics.Start with the idea of humor. This will help you in a big way. It will show that you are engaging with the reader in an entertaining manner. Your essay will then become one that will be read and enjoyed.But there is no reason why you can't get away with talking about another idea entirely. When you have an idea, go with it.The point of this article is to stress that an argumentative essay topic for college students should contain a mix of both traditional and non-traditional topics. As a matter of fact, you have a lot of flexibility when you write your argumentative essay topics for college students.

Kids, Work and Writing to Solve a Problem Essay Topics

Kids, Work and Writing to Solve a Problem Essay Topics Otherwise, then the very first point to know is that you should begin by viewing a range of topics before choosing one that's suitable. Jot down all you know about the subject in a brief list. Your very first job is selecting a topic. Writing a report isn't simple, particularly if you are writing it for the very first time or writing on a new and challenging topic. Many times, you'll need to include at least a brief overview of the ideas in the article you're discussing. The subject of your essay should be relatively simple to write about, but in addition it needs to match the interests of the overall public. Writing an issue and solution essay might appear challenging in the beginning, but as soon as you've got your subject, you're golden. You will find that a number of the topics can be adapted to suit almost any type of writing assignment. The ideal problem solving essay topics are things which you can practically change and they're things which you can write persuasively about. The perfect way to summarize an essay is to start with reading it through a very first time quickly. Permit the plan steer you on how you should compose your essay. The 30-Second Trick for Writing to Solve a Problem Essay Topic s Additionally, when writing an essay, it is likewise required to look at selecting a topic that has a viable or practical solution. Outlining your essay is an excellent means to organize your ideas and flesh out your essay topic. Problem solution essays are a favorite kind of persuasion. The thesis is a crucial area of the introduction of your problem solution essay, but it doesn't mean that you need to compose your paper, for example, thesis, in a strict purchase. The issue and solution essay examples on the page will steer you on the ideal approach which you should take for writing your paper. Remember your essay is about solving problems, thus a solution ought to be a highlight of the essay. If you're going to compose an interesting, unique essay, you will have to do research. The more specific you're, the simpler it is going to be to prove it in your essay. Put simply, the essay should say how you intend to get ready for class. Your essay is all but ready! Since you may see, the problem-solution essay is really simple after you know its basic elements. It's feasible for you to find problem solution essays help if you make an order for your paper from us. An exploratory paper is not uncommon in businesses when they're trying to come across a remedy to an issue and will need to receive all of the potential perspectives and data available. To address a problem through writing isn't as simple as it sounds. From the above mentioned example, you may be clear in regards to the issue solving essay. It's possible for you to discuss more than three in the event the issue is complex. Sometimes you have the issue and you've got to state the issues. Once you get your problem, you will need to speak about it a tiny bit and persuade the reader's opinion that the problem should be resolved. Problem is a question that needs solution. You simply need a couple of problems as remember you don't have a lot of time and you will need to spell out the issues. Or, maybe you're not faced with any particular problem at the present time, but you know you will later on. The only issue is that the 2 people who had the original conversation never meant for anyone else to know about doing it. Another significant problem is the development of internet fraud and hacking.

Essay Samples For Middle School Students - 4 Tips

Essay Samples For Middle School Students - 4 TipsWriting an essay is a difficult task especially for middle school students. The more difficult it gets, the more time you will need to dedicate to research topics and coming up with right words that best suit the content of your essays.Some of the tips below will help you understand the details of essay writing. With these tips, you can make your writing even better.First, you need to prepare the topic you want to discuss. Next, you have to come up with the writing guidelines for this topic. And lastly, you have to decide on the format that will be used to write your essay.First, you need to make sure that you are aware of all the factors which influence the content of your essay. One of the most crucial factors is the topic of your essay. While choosing a topic for writing an essay, you have to be specific and to be aware of the fact that the topic is not the only thing that affects the content of your essay.You also need to know that there are general ones that are linked to the topic and to each other. The other important things are the starting and ending date, the start date and the end date. In addition, you also need to be aware of other factors like your age, class level, race, religion, sex, culture, and even your social class. Thus, when you are writing an essay, you should always consider the list above to make sure that you choose the right topic for your essay.Writing an essay is not easy. It requires long hours of studying and reading. If you are not really confident about the topic and you feel shy, try to prepare the topics before you begin to write.As you can see, most of the things you need to do are just needed by middle school students. However, if you want to be truly confident and productive, you need to take the steps to ensure that you have everything you need to succeed in writing an essay.