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