4 Simple Steps to Writing the Perfect Illinois Essay

4 Simple Steps to Writing the Perfect Illinois Essay To start, lets debunk a popular myth: the perfect essay. Its not real. It cant be, because otherwise wed already havethe essay and there would be no reason to write any more of them! Were not asking for the perfect essay; were asking for an essay that best answers our prompt as it relates to you. This should hopefully alleviate any anxiety you may be feeling, because guess what? Youre an expert on you!If youre still apprehensive, though, lets take a look at the steps you can take to write a strong Illinois admissions essay. Step 1: Brainstorm. Now that youre ready to write, how do you start? How do you begin to compress your experiences into 400 words to best convey your passion? It all starts with brainstorming. Think about the question at the surface: Why do you love your intended major? How do you know? What have you done to demonstrate that love? Now dig a little deeper. What experience first sparked your interest? What memory pulls you back and makes you think, This is what I want to do all the time?Really look to yourself and your past experiences, because those are the memories that make you who you are, and those are the words that will make you memorable. Write all of your thoughts down on paper, see the connections youve made between your life and your major, and choose what stands out most. If youre having trouble thinking of ideas, dont be afraid to follow a tangent. You never know, even the smallest moment could be a winner. Step 2: Plan it out. Now that you have an idea, it’s time to structure it. Make sure you have a unified thought: You want to make a point, not just put a bunch of thoughts on paper. Pinpoint your message and start your outline. It’s a short essay, so you need to be clear. Start with one sentence to pull the reader into the essay. If it helps, think of this intro as one of those musical numbers that begins a flashback in a movie. Next, outline your main points. As youre outlining these points, make sure you continue to answer the prompt. This essay isnt just about you; its about the you whos interested in a certain major. So its important to focus on that part of you, including any related classroom experiences, extracurriculars, job shadowing, volunteering, or part-time or summer jobs. Finally, think of a closing sentence to tie your essay together. Although it can be similar, this shouldn’t be the same as your first sentence. Step 3: Make it you-tiful. After you’ve planned out your first sentence, last sentence, and main points, it’s time to write your essay! (Its also probably time for us to remind you to write your own essay. No one knows you better than you, and counselors can tell when someones essay has been written by someone else.) This isnt your traditional introduction-three-body-paragraphs-conclusion-type essay. Remember, its short, sweet, and focused on you. When you begin writing, every word counts; theres no need for background information or a rendition of your high school transcript. Stay true to you and what you’ve planned out. Bottom line, tell your story. Write about your passion and connect it to your experiences. A personal anecdote about the time you built a robotic fish or felt empowered at a spoken-word open mic is engaging and memorable, exciting, and unique to you. Step 4: Take one last look. Although this step is often overlooked, its often where a good essay becomes great. When youre done writing your essay, step away for awhile and come back to it with fresh eyes. Now reread it from start to finish to ensure that your writing flows well, youve cut out the fluff, youre using active verbs, and no grammatical errors exist. Then have someone elseâ€"perhaps a parent or your favorite teacherâ€"proofread it and offer their feedback, too. Any mistakes may put a stop in an otherwise flowing, engaging story. This is your chance to catch them. Its also your chance to tighten things up and add any finishing touches (like the cherry on top of that ice cream sundae you totally deserve after you hit save one last time!). We hope these tips help you as youre crafting your Illinois essay. Remember, it doesnt have to be perfect; it just needs to relate to you. We cant wait to read what you come up with! applying college planning Illinois Admissions We're here for you as you prepare for college. Whether you're looking for guidance on the college search process or have questions about Illinois, we hope our blogs will help!

Which of the following should your thesis statement achieve

Which of the following should your thesis statement achieve? Question Which of the following should your thesis statement achieve? It should specify the point of the research. All of these are correct. It should provide unity and a sense of direction. It should set the argument and focus the entire paper. Answer ( B ) This question focuses on what a thesis statement should do.   While a thesis statement is a concise statement discussing the central argument of the paper, to maximize its effectiveness, it needs to include other elements as well. Discussion There are several important components to a great thesis sentence: specificity; clarity; stating a position and/or making an argument, and providing focus for the entire paper. Conclusion An effective thesis statement does more than simply introduce the central argument of your paper; it also provides a roadmap for your paper, providing your reader with the specificity, clarity, and focus.

The Lessons That The American Experience With Federalism

Report on the lessons that the American experience with federalism may offer to the development of the European Union Jonathan Martin Brennan – Student ID: 14014998 Politics and Government - Seminar Tutor: Dr Mike Mills Introduction Federalism laid the foundation of the United States of America as the powerful nation we know it today, and therefore it was ideal as an implement in the shaping of the European Union. The concept of federalism has been a part of the US’s political identity for as long as one can remember, but it has also been a key theory in intellectual and philosophical circles in the old continent. In order to outline the lessons that the European Union can learn from the success of federalism in the United†¦show more content†¦Thus, these political divisions are granted a certain degree of autonomy, that unlike in decentralised states, it is based upon constitutional ground, this is, in states that are built by a decentralised system or devolution of powers, the central power has the capacity of overruling any decision it may wish to; however, in a federal state, the withdrawal of policy-making that the central government may not approve of is a much more complex process, as the local authority has the constitutional right to approve its own measures, as sovereignty is constitutionally divided (Norris, 2001. Chapter 7). Effectiveness in multi-level governance (MLG) Alexis de Tocqueville, one of the most important political thinkers of his time, and a great scholar in comparative federalism, in his â€Å"Democracy in America† mentions how â€Å"old Europe† could learn from â€Å"young America† regarding the enactment of the concept of federalism, particularly in terms of â€Å"the problem of legitimacy in the context of multi-level governance† (Nicolaidis Howse, 2001). It is in this aspect, that the EU must avoid its statist approach to sovereignty and must keep on the track of the subsidiarity principle, i.e., â€Å"the allocation or the use of authority within a political order where there is no unitary sovereign† (Nicolaidis Howse, 2001. Chapter 4.) This is, despite the fear of a loss of credit,

Using Photographs and Illustrations to Support Reading Comprehension

Whether they are the cave drawings in the south of France, the cartoons of Hogarth or Satellite pictures, illustrations and photos are powerful ways for students with disabilities, especially difficulty with text, to find and retain information from textbooks and non-fiction. That, after all, is what reading comprehension  is about: understanding and retaining information, and having the ability to retell that information, not performance on multiple choice tests.   Often students with reading difficulties are so stuck I find, when working with struggling readers, that they get so stuck on the code - decoding unfamiliar multi-syllabic words, that they dont get as far as the meaning. More often than not, they actually miss the meaning. Focusing students on text features, such as the illustrations and the captions help students focus on the meaning and authors intent before they actually have to read any text.   Illustrations will help students Understand what the author believes is important in the text.Visualize the context of the non-fiction text (especially history or geography) or content of the chapter/article.  For students who struggle with text, the visual representation of content will help them see the important content.  Learn text specific vocabulary.  An illustration of an insect in a biology text or a plant in a botany text will be accompanied by captions or labels.  Be sure that students note that information in the text.   Using Pictures and Illustrations in Conjunction with Other Text Features An essential part of SQ3R  (Scan, Question, Read, Review, Reread) a long term strategy for developmental reading is to Scan the text.  Scanning basically includes looking over the text and identifying important information. Titles and Subtitles are the first stop on a text walk.  Titles will also help introduce the important topic spedific vocabulary.  Expect a chapter about the Civil War to have specific vocabulary in the subtitles. Be sure to have a list of focus words for flash cards before you start your text walk: Give (or have available) 3 by 5 cards available for students to write down the text specific vocabulary as your do the text walk together.   Captions and Labels accompany most pictures, and should be read as you do the text walk.  Be sure students record all of the important vocabulary, even if they can read them.  Depending on your students sophistication, a picture or a written definition should go on the back.  The purpose should be for your students to be able to define the vocabulary using their own words. The Reading Strategy - The Text Walk The first time you teach the strategy, you will want to walk the child through the whole process.  Later it will be better if you can fade some of your support and have students take more responsibility for the text walk.  This would be a great activity to do in partners across abilities, especially if you have students who benefit from the structure but have stronger reading skills. After reviewing the titles and pictures, have students make predictions:  What will you read about?  What do you want to know more about as you read?  Did you see a picture that surprised you?   Then scan together for vocabulary they should have on their flashcards.  Make a list on the board or using a document on the digital projector in your classroom.

Substance Abuse - 903 Words

Substance Abuse Substance Abuse is a problem for social workers around the country no matter where you go. There are a number of different social problems, and social systems that a social worker will have to deal with when working in this field. I will hope to address the problem of substance abuse, and the different techniques used in order to cure a client suffering from substance abuse. Substance abuse social workers assess and treat individuals with substance abuse problems, which include the abuse of alcohol, tobacco, or other drugs. According to Katherine Von Wormer an expert in substance abuse, social workers treat people with these problems with individual and group therapy, outreach programs, crisis intervention, social†¦show more content†¦There are a number of social work agencies right here in the city that also have great substance abuse programs. For example there is an agency called Drug Detox Now which is located at 404 West 54th Street. This place is run by Dr. Jordan Scher who has been dealing with substance abuse issues for about 30 years. Scher previously opened a drug detox center in Israel that was well renowned. In 1996 he decided to move to the United States and open Drug Detox Now, where he would use many of his treatments from Israel. Another rehab center in the city is the Phoenix House which is located at 164 West 74th. Street. Since 1977 the Phoenix House has been helping people of all ages fulfill a substance free life. The Phoenix House operates more than 30 residential and outpatient programs for adults and adolescents in New York City, and they treating about 2,500 individuals every day. Another substance abuse agency in the city is called INTER-CARE. They are located at 51 East 25th Street. Inter-Care began in 1988 as a two-person clinical practice and is now a multi-disciplinary facility with a staff of twenty full-time employees. As we can see substance abuse is a very demanding field, but it is also a field where y ou can help a number of people. There are a number of different aspects of social work you must master before you attempt to treat a patient of substance abuse. Substance abuse can be a rewarding field where you can make a drastic changeShow MoreRelatedDrug Abuse And Substance Abuse2025 Words   |  9 PagesDrug abuse is the recurrent use of illegal drugs, or the misuse of prescription or over the counter drugs with negative consequences. The two terms, drug abuse or substance abuse can be defined as the use of chemical substances that lead to an increased risk of problems and an inability to control the use of the substance. Although drug and substance abuse do differ from addiction it sometimes can be mistaken for one another. Addiction is a chronic, often relapsing brain disease that causes compulsiveRead MoreSubstance Abuse And The Active Substance Abuser1514 Words   |  7 Pagesmisadventures of the active substance abuser. Unfortunately, many people in this situation may fe el alone and lost when it comes to the pathway to recovery. Not only for the addict or alcoholic, but a course of action that can give back some semblance of control and peace to the spouse. Sadly, the odds are never in favor for either of the two parties involved; however, people determined to salvage their connection with a loved one may yet be able to do so. Substance abuse of a loved one can be a nearRead MoreSubstance Use Disorder And Substance Abuse997 Words   |  4 PagesSubstance use disorder is defined as being a pattern of maladaptive behaviors and reactions brought about by repeated use of a substance, sometimes also including tolerance for the substance and withdrawal reactions. (pg. 294). The individual I will be talking about for the project is someone that came to crave a particular substance and rely on it every day. Their choice to devote so much of their time to their substance caused issues between their family and friends. Both family and friends startedRead M oreSubstance And Substance Abuse Among Inmates2272 Words   |  10 PagesSubstance Abuse among Inmates Substance abuse can be a big problem in our criminal justice system when dealing with the number of inmates who go back to their illegal activity after being released from custody. In specific, inmates who are drug users/ abusers are to be expected to go back to drugs once they get out. Without treatment they do not have the tools, knowledge, or resources to stay clean on the outside when released. The biggest hazard for an ex-inmate in the community is getting backRead MoreThe Problem Of Substance Abuse942 Words   |  4 Pagesuse various assessment tools to determine potential disorders that may be the root to a problem. Though at times a client may seek counseling with a known substance abuse issue there are more cases where a client is oblivious to the potential contribution that substance abuse has on the problem that help is sought for. Understood that substance dependency is nondiscriminatory to age, gender, race or creed, a therapist approaches the matter with the client kno wing that a problem is recognized and thereRead MoreThe Effects Of Substance Abuse1111 Words   |  5 PagesOpiate abuse in the United States is an epidemic. It is argued whether addiction is being handled correctly, and if the amount of resources for combating addiction should be increased. According to the National Institute on Drug Abuse, â€Å"The number of past-year heroin users in the United States nearly doubled between 2005 and 2012, from 380,000 to 670,000† (2014). This nation-wide increase of heroin abusers leads to another issue: how to treat them. One method of treatment is giving them another medicationRead MoreThe Substance Abuse Social Worker1052 Words   |  5 PagesThe Substance Abuse Social Worker Social workers have long worked with people who abuse substances, beginning in the 1800’s with sobriety houses and inebriate asylums, to settlement houses in the profession’s beginnings to developing self-help, consumer-driven programs, designing inpatient detoxification programs, and to applying youth-oriented prevention and education programs. As an addictions counselor, the social worker is a part of a specialized profession that began with the wounded healerRead MoreModels in the Substance Abuse Field 1206 Words   |  5 PagesThe term substance refers to all psycho active substances –â€Å"any substances when taken by a living organism may modify its mood perception, cognition deportment or motor function,† (DSM-IV, 4th edition) whether it is legal or illegal. Substance abuse refers to the utilization of a drug or other substances for non-medical purposes with the aim of engendering a mind-altering effect in the utilizer (MOH, 2005). It involves the utilization of illegal substances such as cannabis, misuse of licit drugsRead MoreSubstance Abuse And Its Effects On Society Essay1579 Words   |  7 PagesAmerica’s silent killer, substance abuse. Substance abuse affects the user and everyone that they are around and most of the time the user doesn’t even realize nor care that it affects other people because they are consumed by the substances. These substances have the power to dramatically change people’s lives, such as the physical toll it takes on our body, the ways in which it corrupts our metal process, and the detrimental effects it has on our social lives. Most substances that are used in our societyRead MoreSubstance Abuse And Drug Therapy1101 Words   |  5 PagesA person who has a substance abuse and a mental health disorder are identified as having a co-occurring disorder. Patients with substance use and mental health share common ground, because they hold the philosophy that treatment of chronic illness requires a long-term approach in which stabilization, education, and self-management are central (Drake, Mercer-McFadden, Mueser, McHugo, Bond, 2016, p. 589). A counselor can help give resources (NA meetings being available in their community) as well

Leadership in Innovation - 4900 Words

LEADERSHIP The activity of leading a group of people or an organization or the ability to do this. Leadership involves (1) establishing a clear vision, (2) sharing that vision with others so that they will follow willingly, (3) providing the information, knowledge and methods to realize that vision, and (4) coordinating and balancing the conflicting interests of all members and stakeholders. Leadership is organizing a group of people to achieve a common goal. ROLE OF LEADERSHIP IN TECHNOLOGY GENERATION Technology is now an integral part of learning and the ability to learn and use technology is quickly becoming a†¦show more content†¦Innovation includes three different stages, which are all dynamic and iterative (constant): 1. Idea Generation 2. Evaluation 3. Implementation The two types of innovation include exploratory innovation, which involves generating brand new ideas, and exploitative innovation, which involves modifying and improving ideas that already exist.[10][11][12][13] It is of importance to note that the ideas generated need to be useful in order to be considered innovative. Innovation should also not be confused with creativity, which is merely the generation of a novel idea that may not necessarily be put into operation, although these words are sometimes used interchangeably in research literature when speaking about innovation leadership. Innovation leadership is a quite complex concept, as there is no single explanation or formula for a leader to follow to increase innovation. As a result, innovation leadership encompasses a variety of different, activities, actions, and behaviors that interact with one another to produce an innovative outcome. Innovative Organizational Culture/Climate Some studies have shown evidence of organizational culture as the mediator of the relationship between transformational leadership and organizational innovation [21][22][23][24] and performance.[25][26] In other words, in order for transformational leadership to have an effectShow MoreRelatedInnovation Leadership2252 Words   |  10 PagesInnovation Leadership MBA6006 Capella University Introduction When one thinks of innovation, it is a term used to describe new ideas and new beginnings. One who introduced this concept to a failing organization, was an innovative leader named. Alan Lafley former CEO of Pamp;G, adapted the five discovery skills that were displayed in Dyer, Gregersen, amp; Christensen ‘s Innovator’s DNA associating, questioning, observing, networkingRead MoreInnovation And Leadership : Leadership Qualities1998 Words   |  8 PagesInnovation and Leadership Brubaker had the foresight and leadership qualities necessary to take on an initiative that would end up serving over 250,000 new members of which 35 percent were uninsured. The social impact Brubaker and her team are a testament to doing the right thing for the right reasons. How did Brubaker make the new program so successful both politically and structurally within Aetna? By use of exemplary communication, a well thought out business plan, and superior leadershipRead MoreThe Role Of Leadership Of A Innovation1549 Words   |  7 PagesThe Role of Leaders in Innovation Innovation is a requirement for any business leader who wants to have sustainable success throughout the global market. Understanding the skills required of a leader in order to create an organization that innovates varies from leader to leader and business to business. However, there are some commonalities, based on the study of organizations and leaders who have been widely accepted for their ability to remain innovative. The following paper will explore viewpointsRead MoreLeadership : Supporting Training Innovation Essay1520 Words   |  7 Pagesâ€Å"Appreciative Leadership: supporting training Innovation† the creator said; â€Å"†¦that thankful administration may be exceptional around authority hypotheses both previous and exhibit. This uniqueness incorporates its strength-based practice, hunt for those certain done kin Also organizations, and the part this assumes Previously, authoritative improvement Furthermore change. † (Knapp Also Copland, 2010). School board policy requires that the faculty support the leadership and the leadership support theRead MoreStrategic Leadership Innovation at Apple4130 Words   |  17 PagesContents Executive Summary 3 Part One – Critically Evaluate the Key Features of Apple’s Industry 5 Part Two – Discuss how Leadership in Apple Influenced Strategic Innovation 8 Part Three – Critically assess Steve Jobs as a strategic leader at Apple 12 Conclusions 15 Appendices 17 References 20    2    Executive Summary â€Å"Increasingly, though, we live in an irregular world, where irregular people take advantage of irregular events and useRead MoreStrategic Leadership Innovation at Apple4142 Words   |  17 PagesContents Executive Summary 3 Part One – Critically Evaluate the Key Features of Apple’s Industry 5 Part Two – Discuss how Leadership in Apple Influenced Strategic Innovation 8 Part Three – Critically assess Steve Jobs as a strategic leader at Apple 12 Conclusions 15 Appendices 17 References 20    2    Executive Summary â€Å"Increasingly, though, we live in an irregular world, where irregular people take advantage of irregular events and use irregular means to produce irregularRead MoreThe Concept of Leadership in Innovation and Change549 Words   |  2 Pages........................... ................................ ................................ ................................ ............ 4 The concept of leadership in innovation and change ................................ ................................ ...... 7 The significance of leaders to the process of innovation ................................ ................................ 9 My reflections ................................ ................................ .......Read MoreInnovation Leadership : 6 Steps Essay1756 Words   |  8 PagesInnovation Leadership: 6 Steps to Profitability By Timothy Donovan | Submitted On March 22, 2011 Recommend Article Article Comments Print Article Share this article on Facebook Share this article on Twitter Share this article on Google+ Share this article on Linkedin Share this article on StumbleUpon Share this article on Delicious Share this article on Digg Share this article on Reddit Share this article on Pinterest Expert Author Timothy Donovan Current Situation In today s competitiveRead MoreLeadership and Innovation at Apple, Inc. Essay595 Words   |  3 Pageslong-term strategy for growth. Also, I will mention what two objectives I believe Apple should include in the learning and growth perspective of their balance scorecard. The first strategic factor that I think Apple should consider is product leadership/ development. Apple became who they are by striving to produce top of the line products that were ahead of the field in technology and electronics (Pearce II Robinson, 2011). Steve Jobs said it best himself, the reason a lot of us are at AppleRead MoreSustainability, Leadership And Innovation : Keys For Success1496 Words   |  6 PagesSustainability, Leadership and Innovation - Keys for Success Throughout this paper I will begin to look at the sustainability that has been put in place for Garmin International and how it affects the current and future leadership of the business. In addition to contributing to sustainable innovation to keep the company relevant and successful in a competitive technology race around the globe. This paper will focus on the keys for success that the company instills into its manufacturing, employees

Probability Exercice Free Essays

MTH3301 Fall 2012 Practice problems Counting 1. A closet contains 6 di? erent pairs of shoes. Five shoes are drawn at random. We will write a custom essay sample on Probability Exercice or any similar topic only for you Order Now What is the probability that at least one pair of shoes is obtained? 2. At a camera factory, an inspector checks 20 cameras and ? nds that three of them need adjustment before they can be shipped. Another employee carelessly mixes the cameras up so that no one knows which is which. Thus, the inspector must recheck the cameras one at a time until he locates all the bad ones. (a) What is the probability that no more than 17 cameras need to be rechecked? b) What is the probability that exactly 17 must be rechecked? 3. We consider permutations of the string †ABACADAFAG†. How many permutations are there? How many of them don’t have any A next to other A? How many of them have at least two A’s next to each other? 4. A monkey is typing random numerical strings of length 7 using the digits 1 through 9 (not 0). Call the digits 1, 2, and 3 †lows†, call the digits 4, 5, and 6 †mids† and digits 7, 8 and 9 †highs†. (a) How many di? er ent strings can he type? (b) How many of these strings have no mids? c) How many of these strings have only one high in them? For example, the string 1111199 has two highs in it. (d) What’s the probability that a string starts with a low and ends with a high? (e) What’s the probability that a string starts with a low or ends with a high? (f) What’s the probability that a string doesn’t have at least one of the digits 1 through 9? 5. School of Probability and Statistics (SPS) at IUA University has 13 male Moroccan professors, 8 female Moroccan professors, and 12 nonMoroccan professors. A committee of 9 professors needs to be appointed for a task. a) How many committees can be made? (b) What’s the probability 1 that the committee contains 2 Moroccan women, 3 Moroccan men, and 4 non-Moroccans? (c) What’s the probability that the committee contains exactly 4 nonMoroccans? (d) What’s the probability that the committee contains at least 4 n onMoroccans? (e) What’s the probability that the committee does not contain any Moroccan men? Conditional Probability, Bayes’ Theorem 1. Before the distribution of certain statistical software every fourth compact disk (CD) is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rate for the 4 testing programs are, respectively, 0. 01, 0. 03, 0. 02 and 0. 01. (a) What is the probability that a CD was tested and failed any test? (b) Given that a CD was tested, what is the probability that it failed program 2 or 3? (c) In a sample of 100, how many CDs would you expect to be rejected? (d) Given a CD was defective, what is the probability that it was tested? 2. A regional telephone company operates three relay stations at di? rent locations. During a one-year period, the number of malfunctions reported by each station and the causes are shown below: Station Problems with electricity supplied Computer malfunction Malfunctioning electrical equipment Caused by other human errors A 2 4 5 7 B 1 3 4 7 C 1 2 2 5 Suppose that a malfunction was reported and it was found to be caused by other human errors. What is the probability that it came from station C? 3. Police p lan to enforce speed limits by using radar traps at 4 di? erent locations within the city limits. The radar traps at each of the locations L1 , L2 , L3 , and L4 are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probabilities 2 0. 2, 0. 1, 0. 5 and 0. 2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket? You can assume that the radar traps operate independently of each other. 4. Jar A contains 6 red balls and 6 blue balls. Jar B contains 4 red balls and 16 green balls. A six-sided die is thrown. If the die falls †6†, a ball is chosen at random from jar A. Otherwise, a ball is chosen from Jar B. If the chosen ball is red, what is the probability that the die fell †6†? 5. The word spelled HUMOR by a person from the United States is spelled HUMOUR by a person from UK. At a party, two-thirds of the guests are from the United States and one-third from UK. A randomly chosen guest writes the word, and a letter is chosen at random from the word as written. (a) If this letter is a U, what is the probability that the guest is from UK? (b) If the letter is an H, what is the probability that the guest is from UK? 6. Jar A contains two black balls, jar B contains two white balls, and jar C contains one ball of each color. A jar is chosen at random. A ball is drawn from the chosen jar and replaced; then again a ball is drawn from that jar and replaced. If both drawings result in black balls, what is the probability that a third drawing from the same jar will also yield a black ball? 7. A jar contains 5 red balls and 10 blue balls. A ball is chosen at random and replaced. Then 10 balls of the same color as the chosen ball are added to the jar. A second ball is now chosen at random and seen to be red. What is the probability that the ? st ball was also red? Discrete Random Variables and their Cumulative Distribution Functions and Probability Mass Functions 1. A dice has 6 sides labelled 1 through 6, and the associated probabilities are a, b, c, d, e, and f respectively. Furthermore, you are told that P ({1, 2, 3}) = P ({4, 5, 6}). This die is tossed once and random variable X is twice the face value that showed up. Answer the following questions about X: 3 (a) What is the range space of X? (b) Draw the cumulative distribution function of X. (c) Write down the probability mass function of X. 2. A jar contains 10 balls, labelled 1 through 10. We will take 3 balls out of the jar. Let B be the random variable that is the highest label among the 3 balls withdrawn. Answer the following questions about B: (a) What is the range space of B? (b) Calculate p(b) for b = 3, 6, 10. (c) Calculate F (b) for b = 3, 6, 10. (d) Calculate P (B ? 8). 3. Consider a group of 5 blood donors, A, B, C, D, E, of whom only A and B have type O+. Blood samples will be taken from each donor in random order, until an O+ donor is reached. Let the random variable Y be the number of blood samples taken until an O+ individual is reached. (a) What is the range space of Y? b) Write down the probability mass function of Y. 4. A jar contains 15 balls, 10 of them red and 5 of them blue. Three balls are picked and let R be the random variable that is the number of red balls in these 3 drawn. (a) What is the range space of R? (b) Write down the prob. mass func. of R. (c) Write down the cumulative distr. func. of R. 5. A random variable Z has following range spac e and probability mass function: 4 value -3 -2. 5 0 4 12 20 probability of this value 0. 1 0. 15 0. 05 0. 3 0. 3 0. 1 (a) Draw the line graph of this PMF. (b) Write down the CDF of Z and draw its graph. (c) Calculate P (Z). . After all students have left the classroom, a probability professor notices that 4 copies of text book were forgotten behind. At the beginning of the next lecture, the professor distributes the 4 books in a completely random fashion to each of the four students who lef the books behind. Let X be the number of students who receive their own book. Determine the pmf of X. Hint: Think of permutations of 4 symbols. 7. Let X be the number of tires on a randomly selected automobile that are underin? ated. Which of the following three p(x) functions is a legitimate pmf for X, and why are the other two not allowed? p(x) p(x) p(x) 0 0. 3 0. 4 0. 4 1 0. 2 0. 1 0. 1 2 0. 1 0. 1 0. 2 3 0. 05 0. 1 0. 1 4 0. 05 0. 3 0. 3 8. In our experiment, we pick a random permutation of 1 234. Let X be the number of symbols that remained in their original places. For example, if the random permutation is 3214, X = 2. Find the pmf of X. 9. In our experiment, we type a random string of length 6 using only the letters A, B, C, D, E, X, Y, Z. Let R be the number of letters that are occuring more than once. So, for example, if the string is †BAYEDA†, R = 1. If string is †DEBAZY†, R = 0. If string is †AABAXY†, R = 1. If string is †AABBXY†, R = 2. (a) How many elements are there in the sample space of the experiment? (b) How many elements in the range space of R? 5 (c) Calculate pR (0). (d) Calculate pR (r) for r ? 4. 8 6 (8)(6)? 7? 6? 5? 4+(8)(6)? 7? 6? 5+(1)(4)? 7? 6+(8)(6)? 7+(8)(6) 1 3 1 5 1 6 . (e) Show that pR (1) = 1 2 6 8 Continuous Random Variables and their Cumulative Distribution Functions and Probability Distribution Functions 1. A college professor never ? nishes his lecture before the bell rings to end the period and always ? nishes his lectures within 2 minutes after the bell rings. Let X equal the time that elapses between the bell and end of the lecture and suppose the pdf of X is f (x) = kx2 0 0? x? 2 otherwise (a) Find the value of k. (b) What is the probability that the lecture ends within one minute of the bell ringing? (c) What is the probability that the lecture continues beyond the bell for between 60 and 90 seconds? (d) What is the probability that the lecture continues for at least 90 seconds beyond the bell? 2. The time X (in minutes) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniform distribution for values of X between 25 and 35. a) Write the pdf of X and sketch its graph. (b) What is the probability that the preparation time exceeds 33 min.? (c) Without computing it mathematically, what do you think is the mean value of X? (d) For any a such that 25 a a + 2 35, what is the probability that preparation time is between a and a + 2 minutes? 3. Consider the following pdf, where k and C are constants: f (x) = kC k xk+1 0 6 x? C x 50 1 625 r If the propolis content is less than 10 parts per thousand, the honey sells for 300 Dirhams per kilogram. If the propolis content is more than 40 parts per thousand, the honey sells for 200 Dirhams per kilogram (because of the too strong taste). If the propolis content is between 10 and 40 parts per thousand, the honey sells for 450 Dirhams a kilogram. Let the price of honey per kilogram be the random variable X. (a) Draw the pdf of R. (b) Determine the pmf of X. Continuous Functions of Continuous Random Variables 1. Let continuous random variable X have pdf fX (x). Let Y = |X|. Write FY (y) in terms of integral(s) of fX . 2. For more practice here, make sure you can do problems number 9 and 13 in Chapter 3 exercises in our text book. 8 How to cite Probability Exercice, Essay examples