...CHAPTER 6 RANDOM VARIABLES PART 1 – Discrete and Continuous Random Variables OBJECTIVE(S): • Students will learn how to use a probability distribution to answer questions about possible values of a random variable. • Students will learn how to calculate the mean and standard deviation of a discrete random variable. • Students will learn how to interpret the mean and standard deviation of a random variable. Random Variable – Probability Distribution - Discrete Random Variable - The probabilities of a probability distribution must satisfy two requirements: a. b. Mean (expected value) of a discrete random variable [pic]= E(X) = = 1. In 2010, there were 1319 games played in the National Hockey League’s regular season. Imagine selecting one of these games at random and then randomly selecting one of the two teams that played in the game. Define the random variable X = number of goals scored by a randomly selected team in a randomly selected game. The table below gives the probability distribution of X: Goals: 0 1 2 3 4 5 6 7 8 9 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001 a. Show that the probability distribution for X is legitimate. b. Make a histogram of the probability distribution. Describe what you see. 0.25 0.20 0.15 0.10 ...
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...STAT200 final exam 2016 Click Link Below To Buy: http://hwcampus.com/shop/stat200-final-exam-2016/ On Multiple Choice questions, please explain and/or show your work as necessary. Other questions please show your work as well. _____1. Ten different senators are randomly selected without replacement, and the number of terms that they have served are recorded. Does this constitute a binomial distribution? Select an answer, and then state why. a. No b. Yes Why: _____2. Which of the following pairs are NOT independent events? a. Flipping a coin and getting a head, then flipping a coin and getting a tail b. Throwing a die and getting a 6, then throwing a die and getting a 5 c. Selecting a red marble from a bag, returning the marble to the bag, then selecting a blue marble d. Drawing a spade from a set of poker cards, setting the card aside, then selecting a diamond from the set of poker cards e. All of the above are independent events _____3. Exam scores from a previous STATS 200 course are normally distributed with a mean of 74 and standard deviation of 2.65. Approximately 95% of its area is within: a. One standard deviation of the mean b. Two standard deviations of the mean c. Three standard deviations of the mean d. Depends on the number of outliers e. Must determine the z-scores first to determine the area _____4. You had no chance to study for the final exam and had to guess for each question. The instructor gave you three choices for the...
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...and B, you i=1 n n=1 should show that ∀x ∈ A, x ∈ B, and vice versa.) (b) Let F be a σ-field on Ω. If An ∈ F for all n ∈ N, then ∩∞ An ∈ F. n=1 3. An elementary school is offering 3 language classes: one in Chinese, one in Japanese, and one in English. These are open to any of the 100 students in the school. There are 28 students in the Chinese class, 26 in the Japanese class, and 16 in the English class. There are 12 students that are in both Chinese and Japanese, 4 that are in both Chinese and English, and 6 that are in both Japanese and English. In addition, there are 2 students taking all 3 classes. (a) If a student is chosen randomly, what is the probability that he or she is not in any of these classes? (b) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class? (c) If 2 students are chosen randomly without replacement, what is the probability that at least one is taking a language class? 4. A CEO wants to decide how much to invest on a project. If the firm invest x, then the probability of success is 1 − e−2x . If the project succeeds, then the firm earns 1, and if not, nothing. Let Y be the earning from the project. The profit function of the firm is given by π(Y |x) = Y − x. (a) Derive the expected profit function, Eπ(Y |x). (b) Suppose that the firm is risk neutral, and therefore wants to maximize the expected profit. How much should they invest? 5. A researcher wants to know how many people have experience in gambling. However...
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...guide. Be sure to show your work in case partial credit is awarded. To receive full credit, work must be shown if applicable. Section 4.1 Probability Distribution 1. Decide whether the random variable is discrete or continuous. (References: example 1 page 195, end of section exercises 13 – 20 page 201) (1 point per each part) a. The cost of a randomly selected apple continuous b. The height of randomly selected Statistics student continuous c. The number of books in the local college library discrete d. The braking time of a motorcycle continuous e. The number of cell phone call between Bagdad and Ft Bragg, NC on Christmas day in 2008 discrete 2. Decide whether the distribution is a probability distribution. If it is not a probability distribution, identify the property that is not satisfied. (References: example 3 and 4 page 197, end of section exercises 25 - 28 page 202 - 203) (5 points) |x |P(x) | |0 |0.49 | |1 |0.05 | |2 |0.32 | |3 |0.07 | |4 |0.07 | Yes, the distribution is a probability distribution because the probability of each value of the discrete random variable is between 0 and 1, inclusive. Also, the sum of all the probabilities is 1. 3. The table shows the probabilities associated with the number of defectives (x) in a group of...
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...2014 1. The following table summarizes results from the sinking of Titanic which had 2,223 passengers: Men Women Boys Girls Survived 332 318 29 27 Died 1360 104 35 18 (a) (2.5 points) If one of the Titanic passengers is randomly selected, find the probability of getting someone who is a girl or boy. (b) (2.5 points) Given that we select a random person who survived, what is the probability of getting boy? (c) (2 points) Suppose that boys who survived, went for cruise at the rate of six per year. Using the Poisson distribution, what would be the mean and standard deviation of the number of boys who survived and went for a cruise in six years? 2. It is found that Viagra users experience headaches with probability 0.15. Suppose we have a group of 105 Viagra users and each of them is independent to experience headaches. (a) (2 points) What is the mean and standard deviation of the number of users that experience headaches? (b) (2 points) Would it be unusual for 19 users to experience headaches? Why or why not? 3. Assume that men’s weight are normally distributed with a mean of 172 and a standard deviation of 29. (a) (3 points) Find the probability that a randomly selected man weights less than 157. (b) (3 points) Find the probability that a randomly selected man weights between 172 and 187. (c) (3 points) Find P20 , which is the weight separating the bottom 20% from the top 80%....
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...Devica Pittiman Lab#2 1) Subset of a statistical population in which each member of the subset has an equal probability of being chosen. A simple random sample is meant to be an unbiased representation of a group. An example of a simple random sample would be a group of 25 employees chosen out of a hat from a company of 250 employees. In this case, the population is all 250 employees, and the sample is random because each employee has an equal chance of being chosen. Sampling error can occur with a simple random sample if the sample doesn't end up accurately reflecting the population it is supposed to represent. For example, in our simple random sample of 25 employees, it would be possible to draw 25 men even if the population consisted of 125 women and 125 men. For this reason, simple random sampling is more commonly used when the researcher knows little about the population. If the researcher knew more, it would be better to use a different sampling technique, such as stratified random sampling, which helps to account for the differences within the population (such as age, race or gender). 2) In a large population it would be in our best interest to use simple random sampling because it would yields unbiased results. As N increases the distribution of P becomes approximately normal. It’s easier for us to use simple random sampling than having to sample ourselves which yields bias results. 3) Advantages are that it is free of classification error, and it requires...
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...Distributions Learning Objectives 1. Understand the concepts of a random variable and a probability distribution. 2. Be able to distinguish between discrete and continuous random variables. 3. Be able to compute and interpret the expected value, variance, and standard deviation for a discrete random variable. 4. Be able to compute and work with probabilities involving a binomial probability distribution. 5. Be able to compute and work with probabilities involving a Poisson probability distribution. A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. Discrete Probability Distributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Distribution n Poisson Distribution [pic] A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. Example: JSL Appliances n Discrete random variable with a finite number of values n Let x = number of TVs sold...
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...determine whether or not my managers views are accurate, that it is impossible to select at random 10 ART cycles that did not result in a clinical pregnancy, several types of equations can and will be used. The equations that were used can be viewed below under Equations 1. The data and results from the equation tell us that it is possible to randomly select 10 ART cycles that produce no pregnancy. While it is possible for there to be 10 ART cycles chosen at random with none of them resulting in a clinical pregnancy, it is highly unlikely. The calculated probability for 10 randomly selected ART cycles that result in no pregnancy is roughly 0.0102. That probability is essentially 0 however it is possible therefore proving my manager incorrect. The number of pregnancies per 10 randomly selected cycles can be best described using a binomial distribution. A binomial distribution is the probability distribution of a binomial variable. The binomial random variable in this case is pregnancy. The binomial trial in this case is whether or not a pregnancy is created from an ART cycle. Each trial can result in two outcomes, either a pregnancy or a non-pregnancy (when factoring ectopic pregnancy as a non-pregnancy). The probability of failure or success for an ART cycle doesn’t change for each trial based on the data given therefore a binomial equation can be used to find the probability of randomly selecting 10 ART cycles with no pregnancies. In this scenario, the binomial probability distribution...
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...Aliyah Shubrick Business Statistics BE220A 3/20/2013 Assignment 7 1. What is a random sample? - Random sampling is a group of subjects randomly selected from a larger group (population) randomly without a bias and influence by the information you are interested in. 2. Why does randomly selecting a number between one and zero help in creating a random sample from a cdf? - A cumulative density function range from zero to one. By selecting a number between zero and one will help create your random sample. Zero to one is the population and if I pick random number like: 0.2, 0.5, and 0.8, these values will become the random sample for a cdf. 3. What is the difference between a discrete and a continuous distribution? -Discrete distribution contains discrete variables where there are infinite numbers of values possible. Such as yes or no questions -Continuous distribution is an infinite probability distribution used to find probability for a continuous range of values. 4. What is the link between a binomial distribution and a hypergeometric distribution? If we were looking at data, when would we expect the data to follow a binomial distribution rather than a hypergeometric distribution? - A binomial distribution is the number of successes and number of games played with replacement. Hypergeometric has the same concept just without replacement. When looking at the number of wins in a baseball game it is better to follow binomial distribution than hypergeometric...
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...Lesson 2: Review of basic concepts of probability theory Coverage: Basic probability rules Random variables and associate concepts Normal distributions Reading: Chapter 2 (1-7), Chapter 3 (1-5, 10-11) and Chapter 4 (1-8) (1 8) Homework: Replicate and complete all the classroom exercises. Print answers of 2.1 (b-c-d), 2.3 (b-c-d) and 2.4(b-c-d) in one (1) page. 1 Business Statistics Lesson 2 - Page 2 Objectives At the end of the lesson, you should be able to: Define and apply the basic probability rules Describe the basic concepts related to random variables D ib th b i t l t dt d i bl Describe and use the properties of means and variances Recognize and understand the most commonly used probability distributions Use the basic data manipulation and descriptive statistical features of SPSS and transfer between SPSS and Excel SPSS, 1 Business Statistics • • • • Review of probability concepts Lesson 2 - Page 3 Probability: is defined on random events (occurrences), takes values between 0 and 1, and can be interpreted as limit of relative frequency (objective probability) Note: In everyday usage, probability might mean the extent of our belief in the occurrence of the event (subjective probability). However, statistics mostly deals with objective interpretation based on relative frequency. j p q y Basic probability rules: P( Sure event) = 1 and P( Impossible event) = 0 P(A or B) = P(A) + P(B) – P(A and B) Consequences: P( not A) = 1 - P(A)...
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...STAT200 final exam 2016 Click Link Below To Buy: http://hwcampus.com/shop/stat200-final-exam-2016/ On Multiple Choice questions, please explain and/or show your work as necessary. Other questions please show your work as well. _____1. Ten different senators are randomly selected without replacement, and the number of terms that they have served are recorded. Does this constitute a binomial distribution? Select an answer, and then state why. a. No b. Yes Why: _____2. Which of the following pairs are NOT independent events? a. Flipping a coin and getting a head, then flipping a coin and getting a tail b. Throwing a die and getting a 6, then throwing a die and getting a 5 c. Selecting a red marble from a bag, returning the marble to the bag, then selecting a blue marble d. Drawing a spade from a set of poker cards, setting the card aside, then selecting a diamond from the set of poker cards e. All of the above are independent events _____3. Exam scores from a previous STATS 200 course are normally distributed with a mean of 74 and standard deviation of 2.65. Approximately 95% of its area is within: a. One standard deviation of the mean b. Two standard deviations of the mean c. Three standard deviations of the mean d. Depends on the number of outliers e. Must determine the z-scores first to determine the area _____4. You had no chance to study for the final exam and had to guess for each question. The instructor gave you three choices for the...
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...46 Probability, Random Variables and Expectations Exercises Exercise 1.1. Prove that E [a + b X ] = a + b E [X ] when X is a continuous random variable. Exercise 1.2. Prove that V [a + b X ] = b 2 V [X ] when X is a continuous random variable. Exercise 1.3. Prove that Cov [a + b X , c + d Y ] = b d Cov [X , Y ] when X and Y are a continuous random variables. Exercise 1.4. Prove that V [a + b X + c Y ] = b 2 V [X ] + c 2 V [Y ] + 2b c Cov [X , Y ] when X and Y are a continuous random variables. ¯ Exercise 1.5. Suppose {X i } is an sequence of random variables. Show that V X = V 2 2 σ where σ is V [X 1 ]. time. 1.4 Expectations and Moments 47 i. Assuming 99% of trades are legitimate, what is the probability that a detected trade is rogue? Explain the intuition behind this result. ii. Is this a useful test? Why or why not? Exercise 1.13. You corporate finance professor uses a few jokes to add levity to his lectures. He is also very busy, and so forgets week to week which jokes were used. i. Assuming he has 12 jokes, what is the probability of 1 repeat across 2 consecutive weeks? ii. What is the probability of hearing 2 of the same jokes in consecutive weeks? iii. What is the probability that all 3 jokes are the same? iv. Assuming the term is 8 weeks long, and they your professor has 96 jokes, what is the probability that there is no repetition across the term? Note: he remembers the jokes he gives in a particular lecture, only forgets across...
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...reference (6684), your surname, other name and signature. Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. Full marks may be obtained for answers to ALL questions. This paper has seven questions. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit. 1. Explain briefly what you understand by (a) a sampling frame, (1) (b) a statistic. (2) 2. The continuous random variable X is uniformly distributed over the interval [–1, 4]. Find (a) P(X < 2.7), (1) (b) E(X), (2) (c) Var (X). (2) 3. Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds...
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...JET Copies: The Decision to Purchase a Backup Copier Businesses are continually faced with difficult and challenging decisions, especially those regarding revenue and whether or not to spend money on situations that may or may not occur, based on speculation or perception. There may be several techniques a business can employ to analyze if perceived investments will produce a desired return and subsequently, profit. Simulation is an analytical process that supports business planning objectives. By utilizing models to support this process, business leaders and managers can forecast potential expenditures and better prepare for roadblocks that can serve as organizational disadvantages. JET Copies is a company facing a dilemma of whether or not it would be cost effective to purchase a second copier in case repair times concerning the original one would result in a loss of revenue. To further analyze their case, I will model the number of days to repair the original copier, the number of weeks between breakdowns, and lost revenue due to breakdowns for a period of 1 year. Additionally, by combining all the modeling components, I will provide a detailed summary to support their decision of whether to invest in the additional copier, and do so with a high level of confidence. I will begin by modeling the number of days to repair. To begin the simulation technique chosen for this case, I first utilized the information communicated by Terri speculating that it will take...
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... 1. Experiment, Outcomes, and Sample space 2. Random Variables 3. Probability Distribution of a Discrete Random Variable 4. The Binomial Probability Distribution 5. The Hypergeometric Probability Distribution 6. The Poisson Probability Distribution 7. Continuous Random Variables 8. The Normal Distribution 9. The Normal Approximation to the Binomial Distribution 2 1 7.10.2015 г. An experiment is a process that, when performed, results in one and only one of many observations. These observations are called the outcomes of the experiment. The collection of all outcomes for an experiment is called a sample space. Table 1 Examples of Experiments, Outcomes, and Sample Spaces Experiment Outcomes Sample Space Toss a coin once Head, Tail S= { Head, Tail} Roll a die once 1, 2, 3, 4, 5, 6 S= {1, 2, 3, 4, 5, 6} Toss a coin twice HH, HT, TH, TT S= { HH, HT, TH, TT} Play lottery Win, Lose S= {Win, Lose} Take a test Pass, Fail S= {Pass, Fail} Select a worker Male, Female S= { Male, Female} 3 A random variable is a variable whose value is determined by the outcome of a random experiment. A random variable that assumes countable values is called a discrete random variable. A random variable that can assume any value contained in one or more intervals is called a continuous random variable. 4 2 7.10.2015 г. Examples of discrete random variables 1. The number of heads obtained in three ...
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