Moment Generating Functions of Random Variables

Moment Generating Functions of Random Variables One way to calculate the mean and variance of a probability distribution is to find the expected values of the random variables X and X2. We use the notation E(X) and E(X2) to denote these expected values. In general, it is difficult to calculate E(X) and E(X2) directly. To get around this difficulty, we use some more advanced mathematical theory and calculus. The end result is something that makes our calculations easier. The strategy for this problem is to define a new function, of a new variable t that is called the moment generating function. This function allows us to calculate moments by simply taking derivatives. Assumptions Before we define the moment generating function, we begin by setting the stage with notation and definitions. We let X be a discrete random variable. This random variable has the probability mass function f(x). The sample space that we are working with will be denoted by S. Rather than calculating the expected value of X, we want to calculate the expected value of an exponential function related to X. If there is a positive real number r such that E(etX) exists and is finite for all t in the interval [-r, r], then we can define the moment generating function of X. Definition The moment generating function is the expected value of the exponential function above. In other words, we say that the moment generating function of X is given by: M(t) E(etX) This expected value is the formula ÃŽ £ etx f (x), where the summation is taken over all x in the sample space S. This can be a finite or infinite sum, depending upon the sample space being used. Properties The moment generating function has many features that connect to other topics in probability and mathematical statistics. Some of its most important features include: The coefficient of etb is the probability that X b.Moment generating functions possess a uniqueness property. If the moment generating functions for two random variables match one another, then the probability mass functions must be the same. In other words, the random variables describe the same probability distribution.Moment generating functions can be used to calculate moments of X. Calculating Moments The last item in the list above explains the name of moment generating functions and also their usefulness. Some advanced mathematics says that under the conditions that we laid out, the derivative of any order of the function M (t) exists for when t 0. Furthermore, in this case, we can change the order of summation and differentiation with respect to t to obtain the following formulas (all summations are over the values of x in the sample space S): M’(t) ÃŽ £ xetx f (x)M’’(t) ÃŽ £ x2etx f (x)M’’’(t) ÃŽ £ x3etx f (x)M(n)’(t) ÃŽ £ xnetx f (x) If we set t 0 in the above formulas, then the etx term becomes e0 1. Thus we obtain formulas for the moments of the random variable X: M’(0) E(X)M’’(0) E(X2)M’’’(0) E(X3)M(n)(0) E(Xn) This means that if the moment generating function exists for a particular random variable, then we can find its mean and its variance in terms of derivatives of the moment generating function. The mean is M’(0), and the variance is M’’(0) – [M’(0)]2. Summary In summary, we had to wade into some pretty high-powered mathematics, so some things were glossed over. Although we must use calculus for the above, in the end, our mathematical work is typically easier than by calculating the moments directly from the definition.

Interview with content editor Harrison Demchick Reedsy

Interview with content editor Harrison Demchick Reedsy â€Å"It’s about creating the very best version of your story†: An interview with Harrison Demchick It’s good to see indies and traditional authors brought together on what really unites them: the story, the craft, and offering great content to readers. And editors have a big part in that, one that is often untold. This is why we like to give them a spot on the Reedsy blog.Today, we interview one of these authors’ unsung heroes: a developmental editor. With 10 years experience both in-house and freelance, Harrison Demchick is one of the great editors we have brought to Reedsy, and one of our very first users. Plus, he has a great story to tell! For the lovers of the written word, I’ve transcribed most of it below.  But for those who want to take part in the discussion, you can directly join us on the hangout!Hi Harrison, good to have you here. You’re â€Å"the world’s most thorough content editor†, according to your Reedsy profile. Did you start out as a content editor or more as a copy editor or proofreader?At the time I started, I didnà ¢â‚¬â„¢t know the distinctions yet. When I was in high school and college I was doing copy-editing for several magazines or newspapers but when I started working in publishing, the publisher just gave me a manuscript and told me to â€Å"edit it†, and for me that inherently involved both the copy and the content.From the very first Summer I began in publishing (2005), I’ve been doing content (developmental) editing. It just took some time before I knew the formal term that was ascribed to it.Do you accept to work with authors who come to you with an unfinished draft? Or is there a particular at which you prefer authors to contact you (first draft, third draft, etc.)?Generally I do prefer working with a finished draft, that makes it much easier to provide overall feedback, especially when it comes to story or character arch, climax, etc. But as far as as polished that draft needs to be, I feel I can be very useful anytime from the first draft on. I personally like going through the whole developmental edit process and provide feedback relatively early, if not right after the first draft, so that the author is able to know what they need to do and have a plan of attack for the next draft.If you start working with an author on the first draft, generally how long does it take for you and the author to reach the final manuscript stage?Well that depends a lot on the context and how long I’m staying with a project. When in traditional publishing I was with a project from start to finish and that could be a process that could last anywhere from 6 months to two years, to make sure that the books that we put out were as strong as they possibly could be.On the freelance side it depends a lot on the author and their direction. I love to stick to projects when I can: I start with a developmental edit and do a smaller consultation afterwards. And if I look at when books happen to be published rather than when I’m finished working on them, it stil l often ends up being more than a year after we start.Obviously it really comes down in the end to how quickly and effectively the author works, and what they want from me afterwards.As you have worked both in-house for a publisher and freelance, do you see any big differences in how you work with authors in both cases? The differences, in my opinion, are not necessarily that vast. Well, the big difference for me as far as what I get to do for a living, is that I now get to focus entirely on the editing, and that’s one of the reasons I chose to go freelance. But as far as the authors, I work with both with authors who plan to self-publish and with authors who plan to start contacting publishers after they’re done working with me.And in either the case the goal is the same: it’s to identify what’s working, what’s not working, and how we can make it better and create the very best version of the author’s manuscript.One could make the case that when working with someone pursuing traditional publishing the focus could be more on how to best market it for publishers. And while I’m happy to give feedback on that, for me it doesn’t matter as far as the story itself is as good as it can be. For me it’s all about creating the best possible version, and that’s the same whether it’s for traditional or self-publishing.I definitely like your point there, it should always be about getting the quality of the writing as high as possible. I also think that some books, according to the genre, are actually more suited for self-publishing (because the target market is smaller, maybe). Do you try to advise authors on which publishing route they should take? Or do you adapt the adapt their writing and their story to the route they’ve chosen?I definitely advise, but I don’t recommend, necessarily. Every author has their own approach and my job is to help them achieve their goals. Of course if I see an issue with those goals or something that could make it easier I will let them know that.Self-publishing and traditional publishing both have their pros and cons, and I don’t see genre as one of them, necessarily, one of the distinguishing characteristics. Generally authors who have decided to self-publish or traditionally publish have done so for specific reasons and as long as those are valid I will do my best to advise them and work in that way. Fortunately, my experience is such that I can advise effectively no matter which direction they choose.Do you think a good editor can work with any author out there, or is there one perfect editor for every author?I certainly wouldn’t go that far. I think there are certainly cases where a particular editor’s personality or approach would work better for a particular author, that is no doubt true. But there are also a lot of disparities in quality among editors out there. A lot of the time it’s not so mu ch searching for the one that fits, than going through a lot of people who are not really fantastic at it before finding somebody who actually is.Fundamentally it comes down to being as effective as you can possibly be. I think that’s   more important than having a particular chemistry with the author you’re working with.I definitely agree with your point on the disparity of quality in editors out there, and that is part of the reason why we created Reedsy.Exactly, and one reason for that is that a lot of people assume that they can edit, because they love to read and they’d love to be helpful to authors. Also, because they don’t know fully what goes into it. And some start with that belief and actually become amazing, which is exciting to see.All this makes it very hard for the authors to know whether the person they’re talking with is someone genuinely skilled or just someone very enthusiastic who wants to believe they are going to be able to h elp the author. That’s why I, too, love what Reedsy is going for because it helps authors navigate that.You are yourself a published author with one book out there: The Listeners, and were also a screenwriter before that. What pushed you to publish? Did editing a lot of books make you want to put your own work out there?Actually, it’s just the opposite. The Listeners started out as a series of short stories I wrote in my last semester of college, around the same time I started my career in editing in a publishing company.This particular series developed in a screenplay, and the publisher I worked for expressed an interest in a novel version, which I wrote. But for a while, my experience in publishing actually made me not want to put the book out there, because I knew the marketing challenges. I knew I would be out there, front and center trying to build a readership, doing book signings, interviews, etc. These are all things that I know how to do, and that I recommend to authors, but things I don’t have a particular knack for myself. I’m not a great marketer or self-promoter, it’s not my nature.Fortunately, I had enough people around me telling me I was an idiot and that I should pursue publishing, so I did, and I’m very grateful for the opportunities I have had since thanks to that.I’d like to finish with an easy question: if you had one recommendation for indie authors out there, whether they’re traditionally published, self-published, or hybrid, what would it be?I suppose it would be to hire me, haha! But more broadly and less egocentrically than that, the thing that divides a writer from an author in my mind is the revision process: pushing forward, doing everything that’s in your power to create the very best version of your story. There is nothing more spectacular than taking an idea not only from concept to first draft but from first draft to final draft. So my advice would be: follow that r oad to the end, do not stop part way through.I agree, and I certainly second that advice. Thanks a lot for your time, Harrison!You can find Harrison Demchick, Ricardo and Reedsy on Twitter: @HDemchick, @RicardoFayet  and @reedsyhqDo you work with a developmental editor? If yes, tell us what he/she brings to your writing in the comments below! And if you have any question for Harrison, do use the same commenting space 🙂

Inspirational Speakers Paper Essay Example | Topics and Well Written Essays - 250 words

Inspirational Speakers Paper - Essay Example g leader who knew that his country was on the brink of greatness and that he would be the person to lead the country to take its rightful place as the leader of the free world. 3. What made his inaugural speech stand out even more for me was the message of his speech. He knew the realities of the new world that our country was a part of after the end of World War 2. He knew that it was time to call upon the people to act. He outlined what he expected of himself as president and how he hoped to achieve it with the help of the citizens of America. His plans were highly ambitious at the time and he knew it. Which is why he called upon every upstanding American citizen to do his part in making the dreams he had for the nation a reality. 4. His style of speech was almost poetic. He used his gift of gab to create an interesting speech that personified the romanticism that would be the defining factor of his presidency. He called upon every American to realize that the country and its past leaders had done everything that they could to help the nation become great and now, the torch had been passed to the younger generation to keep alive. 5. Of all the features of a JFK speech that I might have a chance to emulate within this course would be his (1) convincing way of delivering his speech by using an unwavering voice pattern and (2) his futuristic way of thinking that helps to hone encourage people to dream and make it a reality. 2. I first heard Oprah Winfrey speak during her talk show episode that introduced her unknown half sister to the public. Ms. Winfrey was now a well-known public speaker and inspirational talker who had managed to influence the thoughts and actions of a whole generation of Americans. Ms. Winfrey was in effect, one of the best life coaches that I had ever heard speak. 3. As a life-coach and inspirational talker Ms. Winfrey was concentrated on 2 things. (1.) Knowing the true of value of oneself regardless of public impression and (2.) knowing

Problem Solution Argument Essay Example | Topics and Well Written Essays - 1750 words

Problem Solution Argument - Essay Example The multi-billion dollars academic research funded by marketers is manipulative and unethical so it should be considered illegal. Marketers have the rights to find ways on how to improve their services to maximize their profits. However, there should be a limit to how they employ such ways and maintain ethical principles as they work. Background Information Large companies hire researchers, consultants, and analysts to conduct studies on the various human behaviors using different methods. Especially during global crisis, companies are more driven to increase their sales profits. According to Gladwell article, â€Å"the time per visit that the average American spends in a shopping mall was sixty-six minutes last year down from seventy-two minutes in 1992- and is the lowest number recorded† (Gladwell 95). With economic problems, consumers’ interests of going to malls had fallen. Despite the drop in the number of hours consumers spend in malls, sellers’ profits can be improved through researches on consumer behaviors. Paco Underhill is a researcher who graduated from Columbia who had analyzed hours of shopping videotape in the past decades (Gladwell 95). Through analyzing consumer behaviors, marketers recognize that not all consumers are the same, which is why marketers now do segmentation of consumers. Segmentation is another effective trick and method used by marketers to be able to create the right advertisement for the targeted consumers. Kids are well impacted by the segmentation system. Advertisement has changed rapidly from only a few large companies such as Disney, McDonald, and others, but in our contemporary society, now almost all companies target children for their own benefits. According to Eric Schlosser, â€Å"Today children are being targeted by phone companies, oil companies, and automobilecompanies as well as clothing stores and restaurant chains†. Children are exposed to 40,000 commercials per year. American Companie s are recorded to spend $800 million per year for kids’ advertisement and the revenues are triple. With $800 initial money, they earned $29 billion revenues. Moreover, the Europeans generate revenues between $620 to $930 million. This shows that kids bring huge profits to many producers. Marketers did not only do surveys but they also focused on kids to be questioned secretly without their parents’ notice to obtain the information. Thanks to the Children’s Online Privacy Protection Act that took effect on April, 2000, children’s exposure to abuse are a little limited. Nowadays, clubs seem to be the trend in getting to children. â€Å"According to one Burger King executive, the creation of a Burger King Kids Club in 1991 increased the sales of children's meals as much as 300 percent† (Schlosser). In addition, researchers found out that about 80 percent of kids dream about animals. This information is used by many marketers to create imaginary figure s that will be liked by the young children, used for advertisements to entice them. It is interesting to note how marketers target children because they are easily manipulated due to their immaturity. Lastly, the other technique used by the researchers is Neuromarketing, a process where the possible responses of consumers are studied in order to create products that would meet their ‘

Online School Vs. Traditional Essay Example for Free

Online School Vs. Traditional Essay According to the National Center for Education Statistics, about 4.3 million undergraduate students, or 20 percent of all undergraduates, took at least one distance education course in the 2007-08 school year. And about 0.8 million, or 4 percent of all undergraduates, took their entire program through distance education (May, Ryan). There are many pros and cons about distance and traditional education, as well as many similarities and differences. Online education is a more flexible option for students to get an education since most classes can be taken anywhere you are able to take your laptop or hook up to the internet. It’s very convenient for people who have a busy lifestyle, such as families, full-timer workers, and those who serve in the military. With online learning, students don’t have the traditional methods of sitting at a desk and learning with lectures and books. You have the freedom to learn your way at your own comfort level. Studies have shown that in online learning conditions perform better than those receiving traditional instruction (NCU). However, for some, lectures in the classroom are more beneficial to them since some need more a more hands on approach. A disadvantage of online learning is not having your degree being seen as creditable. Be sure that your college choice is accredited since there are many â€Å"Degree Mills† out looking for your money. There are many students and employers who are skeptics about how much you really do learn from online learning. According to Allie Bidwell from USA news, In a survey of 656 human resources professionals released Wednesday, the researchers found that although nearly half of employers said online-only programs require more discipline, 56 percent said they still prefer applicants with traditional degrees from an average university over those with an online degree from a top university. Overall, most employers (82 percent) said a combination of in-person and online education would benefit the majority of students (Bidwell, 2013). Most employers are going to view how effective an online education is by the quality of the new hires  they bring in. Cost is another consideration to think of when deciding to go online or traditional. Online education is less expensive since there are no extra fees for maintaining a large campus or classrooms. Some classes may not even require books for you to purchase. Either way you choose to go, there are still rules and guidelines that must be followed. Along with that, attendance is still a must with online education and participation in discussions just as you would participate in a classroom. Depending on the type of person you are and the lifestyle you live, either online or traditional education will be right for you. Before getting into either one, students should carefully research both to find out which is the right fit for them. References Bidwell, A. (2013). Employers and Students Remain Skeptical of Online Learning. Retrieved from http://www.usanews.com/news/articles/2013/09/20/employers-students-remain-skeptical-of-online-education North Central University. Online Education Versus Traditional Education. Retrieved from http://www.ncu.edu/why-northcentral/online-learning/learn-more Dudley, S. (2013) Why Online Education Is More Valuable than Traditional College. Retrieved from http:// http://www.wired.com/2013/11/why-online-learning-is-more-valuable-than-traditional-college/ May, R. Traditional Education vs Online Education-Should You Switch? Retrieved from http://www.businessdictionary.com/article/655/traditional-education-vs-online-education-should-you-switch/

Carvers Cathedral Essays -- Essays Papers

Carvers Cathedral Plato’s â€Å"Myth of the Cave† and Carver’s Cathedral provide insight into parallel words. The protagonists in each story are trapped in a world of ignorance because each is comfortable in the dark, and fearful of what knowledge a light might bring. They are reluctant to venture into unfamiliar territory. Fortunately the narrator in the Cathedral is forced by circumstances to take a risk. This risk leads him into new world of insight and understanding. The narrator in â€Å"The Cathedral† begins the story with the issue of hesitation in seeing the light. The light in this story just like the light in Plato’s â€Å"Myth of the Cave† represents reality. The narrator expresses the fear of expressing reality when he said â€Å" I wasn’t enthusiastic about his visit. He was no one I knew. And his being blind bothered me. My idea of blindness came from the movies. In the movies, the blind moved slowly and never laughed. Sometimes they were led by seeing eye-dogs. A blind man in my house was not something I look forward to†. (Page 98). The narrator felt that being blind was like being in a type of prison and the preconceived notion of self-imprisonment was frightening to him. He felt that blindness was exactly like being a prisoner in Plato’s Cave, a scary world where no light ever penetrated. Unfortunately, the husband is imprisoned in his own ignorance. His view of blindness had come from Holly wood’s portrayal of blind people. As far as he is concerned, his situation is completely normal. He knows there are lots of people just like him. In â€Å"The Cathedral† the extent of the husband’s ignorance or naivetà © is extremely irritating. When his wife tells him the beautiful story of the blind man’s romantic relationship with his wife Beulah, all he could think of is â€Å" What a pitiful life this woman must have led. Imagine a woman who could never see herself as she was seen in the eyes of her loved one. A woman who could on day after day and never see the smallest compliment from her beloved. A woman whose husband could never read the expression on her face, be it misery or something better†. (Page 100). But the blind man had sight in the form of intuitiveness. This sight gave him greater vision than the sighted man. The blind man had a sense of and source of reality in the truth and strength of the relationship. This man was unlike the prisoners in the cav... ... when he closes his eyes in order to imagine and draw the cathedral. †Close your eyes now, the blind man said to me. I did it. I closed them just like he said. Are they closed? He said. Don’t fudge. They are closed, I said. Keep them that way, he said. He said, don’t stop now. Draw. So we kept on with it. His fingers rode my fingers as my hand went over the paper. It was nothing else in my life up to now. Then he said, I think that’s it. I think you got it, he said. Take a look. What do you think? But I had my eyes closed. I thought I would keep them that way for a little longer. I thought it was something I ought to do. Well, he said. Are you looking? My eyes were still closed. I was in my house. I knew that. But, I didn’t feel like I was inside anything. It’s really something I said†. (Page 108). By becoming blind he sees clearly how the blind man’s world really is. Being temporarily blind opens his eyes to the world around him. He can understand the handicap, with understanding comes compassion, and the compassion has caused him to develop new insight into the world around him. Interaction with the blind man has allowed him to see, and has removed him from his own personal cave.

Compare and Contrast Lennie and Crooks Essay

Don’t judge a book for its cover. In the book â€Å"Of Mice and Men†, the people in the bunkhouse had their differences and similarities. Crooks and Lennie had their own differences and similarities, Crooks suffered from racism while Lennie didn’t, Crooks was a lonely man while Lennie had George, and both of them were outside of society. Crooks suffer from racism. He lives alone in his own bunkhouse. Curley’s wife treats him like if he was her slave, one time she entered his room and started denigrating him because of his color. He also doesn’t goes to the â€Å"saloon† to play cards or hang out with the other guys. Lennie has George; they have someone to count on. Tough George sometimes makes fun of Lennie they still know someone is there. On the other hand, Crooks has nobody to rely on. Also, they don’t go further their relationship. They both remain outside of society. Crooks started being a hostile; he wanted to be alone and didn’t let anyone get into his room. Lennie is mentally retarded; this is what makes other people make fun of him or he’s unable to do things. At last, they both rest in Crooks room because they don’t fit in the group of men. As they weren’t the same they still had their own differences and similarities, which made them, get together. Lennie had George but Lennie didn’t fit in the group that made him lonely and that’s one part that brings them together, by loneliness. People are not meant to be alone, everyone is meant to have someone in their lives.