How to Calculate the Variance of a Poisson Distribution

How to Calculate the Variance of a Poisson Distribution The variance of a distribution of a random variable is an important feature. This number indicates the spread of a distribution, and it is found by squaring the standard deviation. One commonly used discrete distribution is that of the Poisson distribution. We will see how to calculate the variance of the Poisson distribution with parameter ÃŽ ». The Poisson Distribution Poisson distributions are used when we have a continuum of some sort and are counting discrete changes within this continuum. This occurs when we consider the number of people who arrive at a movie ticket counter in the course of an hour, keep track of the number of cars traveling through an intersection with a four-way stop or count the number of flaws occurring in a length of wire. If we make a few clarifying assumptions in these scenarios, then these situations match the conditions for a Poisson process. We then say that the random variable, which counts the number of changes, has a Poisson distribution. The Poisson distribution actually refers to an infinite family of distributions. These distributions come equipped with a single parameter ÃŽ ». The parameter is a positive real number that is closely related to the expected number of changes observed in the continuum. Furthermore, we will see that this parameter is equal to not only the mean of the distribution but also the variance of the distribution. The probability mass function for a Poisson distribution is given by: f(x) (ÃŽ »x e-ÃŽ »)/x! In this expression, the letter e is a number and is the mathematical constant with a value approximately equal to 2.718281828. The variable x can be any nonnegative integer. Calculating the Variance To calculate the mean of a Poisson distribution, we use this distributions moment generating function. We see that: M( t ) E[etX] ÃŽ £ etXf( x) ÃŽ £etX ÃŽ »x e-ÃŽ »)/x! We now recall the Maclaurin series for eu. Since any derivative of the function eu is eu, all of these derivatives evaluated at zero give us 1. The result is the series eu ÃŽ £ un/n!. By use of the Maclaurin series for eu, we can express the moment generating function not as a series, but in a closed form. We combine all terms with the exponent of x. Thus M(t) eÃŽ »(et - 1). We now find the variance by taking the second derivative of M and evaluating this at zero. Since M’(t) ÃŽ »etM(t), we use the product rule to calculate the second derivative: M’’(t)ÃŽ »2e2tM’(t) ÃŽ »etM(t) We evaluate this at zero and find that M’’(0) ÃŽ »2 ÃŽ ». We then use the fact that M’(0) ÃŽ » to calculate the variance. Var(X) ÃŽ »2 ÃŽ » – (ÃŽ »)2 ÃŽ ». This shows that the parameter ÃŽ » is not only the mean of the Poisson distribution but is also its variance.

Virgie Ammons Patents a Fireplace Damper Actuating Tool

Virgie Ammons Patents a Fireplace Damper Actuating Tool Virgie Ammons was an inventor and woman of color who invented a device for dampening fireplaces. She received a patent for a fireplace damper actuating tool on September 30, 1975. Little is known about the life of Virgie Ammons. One source says she was born on Dec. 29, 1908 in Gaithersburg, Maryland and died on July 12, 2000. She lived in West Virginia for most of her life.  Ammons filed her patent on August 6, 1974, at which time she was living in Eglon, West Virginia. There is no information to be found about her education, training or profession. One unverified source says she was a self-employed caretaker and a practicing Muslim who attended services in Temple Hills. Fireplace Damper Actuating Tool - Patent US 3,908,633 A fireplace damper actuating tool is a tool that is used to open and close the damper on a fireplace. It keeps the damper from opening or fluttering in the wind.   If you have a fireplace or stove, you may be familiar with the sound of a fluttering damper. A damper is an adjustable plate that fits in the flue of a stove or the chimney of a fireplace. It helps control the draft into the stove or fireplace. Dampers could be a plate that slides across the air opening, or it could be fixed in place in the pipe or flue and turned so the angle allows more or less air flow. In the days when cooking was done on a stove that was powered by burning wood or coal, adjusting the flue was a way of controlling the temperature. Virgie Ammons may be have been familiar with these stoves, given her date of birth. She may also have lived in an area where electric or gas stoves were not common until later in her life. We have no details as to what her inspiration was for the fireplace damper actuating tool. With a fireplace, opening the damper allows more air to be drawn into the fireplace from the room and convey the heat up the chimney. More air flow can often result in more flames, but also in losing more heat rather than warming the room. Keeping the Damper Closed The patent abstract  says Ammons damper actuating tool addressed the problem of fireplace dampers that flutter and make noise when gusty winds affected the chimney Some dampers do not remain fully shut because they have to be light enough in weight so the operating lever can open them easily. This makes small differences in air pressure between the room and the upper chimney draw them open. She was concerned that even a slightly open damper could cause a significant loss of heat in winter, and could even result in loss of coolness in summer. Both would be a waste of energy. Her actuating tool allowed the damper to be closed and held closed. She noted that when not in use, the tool could be stored next to the fireplace. No information was found as to whether her tool was manufactured and marketed.

How Apple Does It (Time Magazine Oct 24 2005) Essay

How Apple Does It (Time Magazine Oct 24 2005) - Essay Example 1 According to the conventional wisdom, described in the article, the Apple is operating like closed mini economy, and thus the company might be doomed as it attempts to do everything at once. Apple produced hardware, operating system for it and programs; traditional approach would allow Apple to license some of its products to other companies that specialize in the development of the products and then access the products produced by the companies mentioned. However in my opinion, the company would really increase the efficiency and competitiveness of its products if it followed more traditional approach as in this case, new innovations and consequently new products might appear much quicker, then the current policy adopted by the company. Huge diversification in one company might erode specialization, which is the cornerstone of innovations; it certainly does mean that differentiation and strategy adopted by Apple might not be successful in some circumstances, as the example of ipod clearly indicates, yet in my opinion it was rather the exception from the rule rather than the result of it. In order to access the effectiveness of the strategy implemented by the company, one should understand the princ... In spite of the fact that this product is just several years old, none of the competitors of Apple so far has been able to provide its customers with identical products of digital music industry. So the price of the Ipod certainly reflects some unique added features related to this product. Another element of competitive advantage strategy is the differentiation focus, when the company attempts to differentiate within the segment of the targeted audience. In this case the company should provide the customers with the product that matches the needs of the customers in the situation when current products of the competitors may not meet the needs of the targeted customers. All this features are present in this case as ipod has many unique features not available in the products of the competitors. Let us evaluate the this product within five forces model designed by Michael Porter that comprises the threat of new competitors, the threat of entry in the market, the threat of substitutes, the threat of bargaining power of suppliers and buyers.3 The threat of competitors. Not strong. Sony has started providing the customers with its new product- mini version of Walkman at the end of 2004. According to the research conducted by Moseberg, journalist from the Wall Street Journal, new mini player of Sony was thinner and wider, thus design was more convenient in use, as well as the battery life of the product was much longer. 4 However, when the Sony products appeared in the market the ipod was still unbeatable in the speed with which MP3 songs could be transferred to the player; whereas it took 2 hours and 13 minutes to transfer the 416 to the player of Sony, it took less than five minutes to

The Highest Quality of Under Armour Case Study Example | Topics and Well Written Essays - 500 words - 95

The Highest Quality of Under Armour - Case Study Example It is not well established as compared to the established competitors like Nike, Puma, and Adidas, Under Armour is gain, ng lead in the industry. The function is in terms of the products abilities to conform to situations and perform as per the expectations. Since the company concentrates on the American domestic market, the information necessary to pinpoint performance is timely. Domestic sales are an indicator that Under Armour has a small sales volume. Apart from using the function system to pinpoint performance results, the company can use the area and unit system to pinpoint performance results. Benchmarking is essential to any organization (Baiman and Joel pp. 201). While aware of the existing competitors who include Nike, Reebok, Puma, and Adidas among others, the Under Armour, through benchmarking has managed to evaluate its performance. An issue of concern is its high dependence on few third-party suppliers. The competitors rely on a variety of third party providers thus easy to counter pricing and ensure efficient product distribution all over the market segments. Under Armour suffers massive losses because of relying on few suppliers who end up demanding high prices. Should the demand for suppliers’ products exceed what the company supplies, it can end up affecting its operations due to reduced access to supplies and raw materials. A key measure and standard that Under Amour upholds is obeying laws and regulations. It is an important element in its code of ethics. Launching of a new product in market demands Under Armour to observe standards of identifiability, accessibility, substantiality, and measurability. It is an indicator to show the company’s ability to assess response patterns and incorporate marketing mix in its standards. Trough this information, Under Armour can segment its market and strive to attain a competitive advantage. The strategy is in line with its recommended strategic plan, which has a passionate taglineâ€Å".

BUS599 MoD 1 Case Assign Essay Example | Topics and Well Written Essays - 1500 words

BUS599 MoD 1 Case Assign - Essay Example Whole Foods Market tangible resources are categorized into the three dimensions: organizational resources, the financial resources, physical resources, and technological resources. Its core values include selling of the highest organic and natural products, satisfaction of the customers, support of the team member excellence and happiness, caring about the environment and our communities, and creation of wealth through growth and profits. Through the core values, the company has developed aids through core competencies, which give them a competitive advantage among its competitors. Whole Food Market’s strategy goals is based on supply and demand and placement. The company is actively determined to meet supply and demand by investing a great amount of resources into political reform in regard to organic farming, carrying out R&D to provide high quality products & services, & engagement in outsourcing to ensure efficient production of its products and services (Harasta and Hoffman, 2007). On the side, placement acts as a guidance for setting strategic goals such as; development of the supportive nutritional, investment in the sales and financial research, etc. (wholefoodsmarket.com, n.d.: 2011) Based on the financial reports from the fiscal years, 2005 to date, the profit margin for Whole Food Market has been generally decreasing (United Natural Foods, 2011; UNFI, 2011). The decrease has been largely attributed to the rising costs of goods and increased debts. In addition, compared to its competitors, Whole Foods Market has recorded low EPS and extremely low revenues (McLindon, et al., 2012). According to Yahoo Financials, Whole Foods Market capitalization has increased. The marketing of Whole Foods Market is not that effective as compared to its competitors hence necessitating the need for R&D to help it maintain a steady and upward market capitalization, which is essential in

Apple's Balance Sheet Essay Example | Topics and Well Written Essays - 750 words

Apple's Balance Sheet - Essay Example Assets From the provided financial information for Apple for the two six months operating periods taken in 2008 and 2009, it is clear that the company experienced an upward movement in different items regarding assets. Initial cash and its equivalents for the two specified periods represent a significant level of reduction from USD 11.875b in 2008 to USD 4.466b in 2009 (Apple, 2009). Whereas cash and similar cash items growth could not necessarily represent growth in view of liquidity practicality issues, it is appropriate to make the assumption that the materiality of the difference did not affect the overall reflection of business operations as cash flow values indicate. This argument is also supported by the observation of increased short-term marketable securities over the same period by double from USD 10.236b to USD 20.547b which could explain the reflected liquidity difference. Despite the fact that the accounts receivable after relevant reconciliation adjustments give a reduc tion in 2009, the difference does indicate a huge difference in terms of operations. Held inventories also indicate a reduction that would be explained by operations at a time when demand projections would be scaled downwards amidst financial crisis. However, overall liquid assets reflected a growth of USD 1.542b (from USD 32.311b to USD 33.853b), which gives a better reflection that the company grew its liquid assets. Investment in form of long-term securities gives detail of fixed assets and the increase from USD 2.379b to USD 3.865b (37 per cent) confirms the current assets trend as a growth trajectory (Apple, 2009). Other fixed assets elements such as buildings and properties owned increased from USD 2.546b to USD 2.455b, a reflection also reflected in the category of other assets that rose from USD 2.498b to USD 1.935b. To confirm this trend, the overall asset figure was a growth of 9.3 percent (USD 43.237b from USD 39.572b). In support of this information, perhaps the company can consult certain incidental information of cash flow nature to highlight the apparent growth in assets and operation level. The value of income and other operating activities indicate upward movements in net income (USD 2.81b from USD 2.626b) as was the reconciliation movements for income generating activities. Certain operating concepts such as depreciation and amortization illustrate a rise in the charged values, which translates in growth of the associated assets. An increase in the value of these operation costs for instance for depreciation from USD 222m in 2008 to USD 330m a year later reflects a huge hike of related asset base to support operations of the corporation. In other movements, increased stock-based compensations cost by 45 per cent (USD 242m to USD 351m) is an indication of growth in operations that occasion stock-based compensations, translated as growth by the company. It is evident that this information reflects a powerful suitability of Apple as an investmen t destination. Despite the fact that such increase in compensation could also indicate inefficiency in operations at the same operating capacity, the reduction of disposition losses from USD 10m in 2008 to USD 8m in 2009 as well

Research proposal Paper Example | Topics and Well Written Essays - 500 words - 2

Proposal - Research Paper Example This research proposal aims at establishing the impact that the use of social media as a means of communication has had on human conduct and behavior. A number of issues have been presented in favour of continued use of social media as a means of communication. Some of the arguments include quick means of information delivery, promotion of business operations, aiding learning in schools among other views. Arguments against the use of social media include loss of privacy, identity theft, loss of face to face interaction among others. However there is less information that can be used to defend either stand on the impact of social media on human beings and this forms the basis for this research proposal. The methods that will be applied in data collection will be based on both secondary and primary sources. This will include the use of questionnaires with open end questions presented to students who use social media as a means of communication, personal interviews aimed at receiving direct feedback from various categories of people and online research on the topic. The statistical analysis will take into consideration the variation between the qualitative and quantitative data. Quantitative data will be done by computing the mean and standard deviation of all the values. Correlation coefficients between human behavior and usage of social media will be taken into consideration. Linear regression will also be undertaken when analyzing the specific cases where social media influences the behavior of human beings. Analysis of the content from in depth interviews will form part of the qualitative