Probability Distribution Functions
I summarize here some of the more common distributions utilized in probability and statistics. Some are more consequential than others, and not all of them are utilized in all fields.For each distribution, I give the denomination of the distribution along with one or two parameters and betoken whether it is a discrete distribution or a perpetual one. Then I describe an example interpretation for a desultory variable X having that distribution. Each discrete distribution is tenacious by a probability mass function f which gives the probabilities for the sundry outcomes, so that f(x) = P (X=x), the probability that an arbitrary variable X with that distribution takes on the value x. Each perpetual distribution is tenacious by a probability density function f, which, when integrated from a to b gives you the probability P (a ≤ X ≤ b). Next, I list the mean µ = E(X) and variance σ2 = E((X −µ)2) = E(X2)−µ2 for the distribution, and for most of the distributions I include the moment engendering function m(t) = E(Xt). Finally, I denote how some of the distributions may be utilized.
1)GAUSSIAN (NORMAL DISTRIBUTION)
The Gaussian distribution—a function that tells the probability that any real observation will fall between any two real limits or real numbers, as the curve approaches zero on either side. It is a very commonly occurring continuous probability distribution. In theory, Gaussian distributions are extremely important in statistics and are often used in the natural and social sciences for real-valued random variables whose distributions are not known. Gaussian distributions are also sometimes referred as Bell curve or normal distribution.
PROBABILITY DENSITY FUNCTIONS OF TH...
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...e probability of an arbitrary variable withGaussiandistribution of mean 0 and variance 1/2 falling in the range ; that is
These integrals cannot be expressed in terms of elementary functions, and are often verbally expressed to be special functions *. They are proximately cognate, namely
For a generic normal distribution f with mean μ and deviation σ, the cumulative distribution function is
The complement of the standard normal CDF, , is often called the Q-function, especially in engineering texts. It gives the probability that the value of a standardGaussiandesultory variable X will exceed x. Other definitions of the Q-function, all of which are simple transformations of , are withal used infrequently.
The graph of the standard normal CDF has 2-fold rotational symmetry around the point (0,1/2); that is, . Its ant derivative (indefinite integral) is .
could vary by as much as 4%. Assumption of SD of 2% may be not be correct.
Renaud, R. (2014a, April 10). Unit 10 - Understanding Statistical Inferences [PowerPoint slides]. Retrieved from the University of Manitoba EDUA-5800-D01 online course materials.
A researcher determines that 42.7% of all downtown office buildings have ventilation problems. Is this a statistic or a parameter; explain your answer.
In the play Doubt, by John Patrick Shanly, Sister Aloysius is treating Father Flynn unfairly. Sister Aloysius is the principal of St. Nichols School, who is suspicious and always doubt everyone, especially Father Flynn. She thinks that Father Flynn is guilty, but has no proof. Sister Aloysius doesn’t like Father Flynn in the school and his ideas. She treats him unfairly. Sister Aloysius treats Father Flynn unfairly when she still accuses Father Flynn of giving the altar wine to Donald Muller after Father Flynn tells her the truth. She treats him unfairly by forcing him to request the transfer without proving if Father Flynn is guilty or not and also makes him resign by lying about his past.
In “The Fish” by Elizabeth Bishop, the narrator attempts to understand the relationship between humans and nature and finds herself concluding that they are intertwined due to humans’ underlying need to take away from nature, whether through the act of poetic imagination or through the exploitation and contamination of nature. Bishop’s view of nature changes from one where it is an unknown, mysterious, and fearful presence that is antagonistic, to one that characterizes nature as being resilient when faced against harm and often victimized by people. Mary Oliver’s poem also titled “The Fish” offers a response to Bishop’s idea that people are harming nature, by providing another reason as to why people are harming nature, which is due to how people are unable to view nature as something that exists and goes beyond the purpose of serving human needs and offers a different interpretation of the relationship between man and nature. Oliver believes that nature serves as subsidence for humans, both physically and spiritually. Unlike Bishop who finds peace through understanding her role in nature’s plight and acceptance at the merging between the natural and human worlds, Oliver finds that through the literal act of consuming nature can she obtain a form of empowerment that allows her to become one with nature.
Solving a mystery is not the same as deducing from first principles. Nor does it amount simply to collecting a number of particular data from which to infer a general law. It means, rather, facing one or two or three particular data apparently with nothing in common, and trying to imagine whether they could represent so many instances of general law you don’t yet know, and which perhaps has neve...
During my time producing lemonade at Colt’s Lemonade Stand I sold a lot of cups at my lemonade stand. My mean for cups sold at 27.48, the median was 22, the mode was 20, 28 and the range was 52. On day 4 I sold 52 cups because it was 94 degrees during that time we sold a lot because it was really hot.
“Marginal analysis involves changing the value(s) of the choice variable(s) by a small amount to see if the objective function can be further increased (in the case of maximization problems) or further decreased (in the case of minimization problems)” (Thomas & Maurice, 2012, pp. 91). Marginal analysis is known as “the central organizing principle of economic theory” for its importance and applicability to many aspects of our daily lives as well as our careers (Thomas & Maurice, 2012, pp. 94). The key concepts of marginal analysis include total benefit, total cost, marginal benefit, marginal cost and net benefit. These concepts all come together to play a significant role in the use of marginal analysis to reach the optimal desired outcome.
The x-bar is the symbol used to represent the sample mean, a statistic, and that mean is used to estimate the true population parameter, mu.
Social problem is a broad topic, there is “No conclusive idea of what constitutes a social problem.” To define a social problem, there are generally three different ideas to define a social problem, “Something that impacts a large group; Something that the people in a society collective agree it is problematic; Something that violates a moral code.” (Logan) Healthcare has been on the spot light, because of The American Health Care Act. I’d like to present health care in United States as a social problem, because it qualify the three ideas to define social problem. First of all, it impacts a large group in the society, because of its cost. According to CDC, “28.2 million people who are under age of sixty five are insured” (CDC). Second, people in a society collective
Standard scores are as well call z-values, z-scores, normal scores; the make use of Z be since the normal distribution be too known while the Z distribution.
Income inequality continues to increase in today’s world, especially in the United States. Income inequality means the unequal distribution between individuals’ assets, wealth, or income. In the Twilight of the Elites, Christopher Hayes, a liberal journalist, states the inequality gap between the rich and the poor are increasing widening, and there need to have things done - tax the rich, provide better education - in order to shortening the inequality gap. America is a meritocratic country, which means that everybody has equal opportunity to be successful regardless of their class privileges or wealth. However, equality of opportunity does not equal equality of outcomes. People are having more opportunities to find a better job, but their incomes are a lot less compared to the top ten percent rich people. In this way, the poor people will never climb up the ladder to high status and become millionaires. Therefore, the government needs to increase all the tax rates on rich people in order to reduce income inequality.
There have been many great mathematicians in the world, though many are not well known. People have been studying math for ages, the oldest mathematical object dated all the way back to around 35,000 BC. There are still mathematicians today, studying math and figuring out ways to improve the mathematical world. Some of the most well-known mathematicians include Isaac Newton, Albert Einstein, and Aristotle. These mathematicians (and many more) have influenced the mathematical world and mathematics would not be where it is today without them. There were many great individuals who contributed greatly in mathematics but there was one family with eight great mathematicians who were very influential in mathematics. This was the Bernoulli family. The Bernoulli family contributed a lot to mathematics, medicine, physics, and other areas. Even though they were great mathematicians, there was also hatred and jealousy between many of them. These men did not want their brothers or sons outdoing them in mathematics. Most Bernoulli fathers told their sons not to study mathematics even if they wanted. They were told to study medicine, business, or law, instead, though most of them found a way to study mathematics. The mathematicians in this family include Jacob, Johann, Daniel, Nicolaus I, Nicolaus II, Johann II, Johann III, and Jacob II Bernoulli.
Median-(McMillan, 2012 p.128) the book describes as the middle score of a distribution. As an example, I would use the median as the midpoint while interpreting data measurements.
b. Statistical Modeling: Decide about which distribution pattern a data set follows. It eases the descriptive analysis and provides foundation for predictive analysis.