...VALUE AND VARIANCE Expected Value Variance 5.4 BIVARIATE DISTRIBUTIONS, COVARIANCE, AND FINANCIAL PORTFOLIOS A Bivariate Empirical Discrete Probability Distribution Financial Applications Summary 5.5 BINOMIAL PROBABILITY DISTRIBUTION A Binomial Experiment Martin Clothing Store Problem Using Tables of Binomial Probabilities Expected Value and Variance for the Binomial Distribution POISSON PROBABILITY DISTRIBUTION An Example Involving Time Intervals An Example Involving Length or Distance Intervals HYPERGEOMETRIC PROBABILITY DISTRIBUTION 5 5.6 5.7 74537_05_ch05_p215-264.qxd 10/8/12 4:05 PM Page 219 5.1 Random Variables 219 Exercises Methods SELF test 1. Consider the experiment of tossing a coin twice. a. List the experimental outcomes. b. Define a random variable that represents the number of heads occurring on the two tosses. c. Show what value the random variable would assume for each of the experimental outcomes. d. Is this random variable discrete or continuous? 2. Consider the experiment of a worker assembling a product. a. Define a random variable that represents the time in minutes required to assemble the product. b. What values may the random variable assume? c. Is the random variable discrete or continuous? Applications SELF test 3. Three students scheduled interviews for summer employment at the Brookwood Institute. In each case the interview results in either an offer for a position or no offer. Experimental outcomes are...
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...| Z score (standardised value)-how many sds from the mean the value liesZ score = data value – mean Standard deviation | Metric Data = ExploreCategory = Frequencies | Bigger sample size will give a narrower confidence interval range (more specific) outliers affect the mean but not the median – this is why the median is preferred here. | | Reports -Only give confidence interval if signicant-All values to 2 dec pts except the p-value Experimental = IV is manipulated to see the effect on the DV Observational = Information just observed & recorded | P-Value Significant Figurep-value < 0.05 = Significantp-value < 0.05 = Not Significant The probability that our test statistic takes the observed value Always leave at 3 decimal places | Levene’s Test-Used to test if equal variancesIf significant (<0.05)– use equal variances not assumed rowIf not significant (>0.05)– use equal variances assumed row | Dependent Variable = the variable in which we expect to see a changeIndependent Variable = The variable which we expect to have an effect on the dependent variable Example: There will be a statistically significant difference in graduation rates of at-risk high-school seniors who participate in an intensive study program as opposed to at-risk high-school seniors who do not participate in the intensive study program." (LaFountain & Bartos, 2002, p. 57)IV: Participation in intensive study program. DV: Graduation rates. | Nuisance variable - Associated...
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...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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...ROYAL UNIVERSITY OF PHNOM PENH Master of IT Engineering PROBABILITY AND RANDOM PROCESSES FOR ENGINEERING ASSIGNMENT Topic: BASIC RANDOM PROCESS Group Member: 1, Chor Sophea 2, Lun Sokhemara 3, Phourn Hourheng 4, Chea Daly | Academic year: 2014-2015 I. Introduction Most of the time many systems are best studied using the concept of random variables where the outcome of random experiment was associated with some numerical value. And now there are many more systems are best studied using the concept of multiple random variables where the outcome of a random experiment was associated with multiple numerical values. Here we study random processes where the outcome of a random experiment is associated with a function of time [1]. Random processes are also called stochastic processes. For example, we might study the output of a digital filter being fed by some random signal. In that case, the filter output is described by observing the output waveform at random times. Figure 1.1 The sequence of events leading to assigning a time function x(t) to the outcome of a random experiment Thus a random process assigns a random function of time as the outcome of a random experiment. Figure 1.1 graphically shows the sequence of events leading to assigning a function of time to the outcome of a random experiment. First...
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...sier!™ ing Everything Ea Mak ta t i s t i c s S e nt ia l s Ess Learn: • Exactly what you need to know about statistical ideas and techniques • The “must-know” formulas and calculations • Core topics in quick, focused lessons Deborah Rumsey, PhD Auxiliary Professor and Statistics Education Specialist, The Ohio State University Statistics Essentials FOR DUMmIES ‰ by Deborah Rumsey, PhD Statistics Essentials For Dummies® Published by Wiley Publishing, Inc. 111 River St. Hoboken, NJ 07030-5774 www.wiley.com Copyright © 2010 by Wiley Publishing, Inc., Indianapolis, Indiana Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for the Rest...
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...and for which the techniques of this chapter are useful: Manufacturing 1. Dollar value of inventory 2. Percent defective in manufacturing process 3. Number of days to receive shipments 4. Number of overtime hours per week 5. Hours of labor per unit 6. Weights of manufactured parts Marketing: 1. Ages of credit card holders 2. Numerical ratings of store performance by customers 3. Monthly market share 4. % of customers lost per month 5. Miles travelled by customers to store Finance: 1. Interest rates paid as deviation from prime rate 2. Company’s stock price as % of Dow Jones 3. Ratios of annual dividends to earnings 4. Several companies’ price/earnings ratios Accounting: 1. Accounts receivable 2. Accounts payable 3. Cash balance 4. Value of inventory Personnel: 1. Time with company 2. Years of education 3. Score on company entrance test 4. Job performance rates 5. Annual salary 6. Sick days taken per year Quality Control: 1. Time between defects 2. Defective items per shift 3. % out of tolerance 1 Statistical Analysis for Managers Week 3 4. Time until part failure 5. Time until defective condition detected The Binomial Distribution The binomial distribution is widely applicable in business because so many situations involve the concept of success versus failure. Some examples of situations that can be modeled with the binomial distribution are: 1. Manufactured or purchased parts either meet quality control specifications or don’t. 2. A factory either...
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...DESCRIPTIVE STATISTICS & PROBABILITY THEORY 1. Consider the following data: 1, 7, 3, 3, 6, 4 the mean and median for this data are a. 4 and 3 b. 4.8 and 3 c. 4.8 and 3 1/2 d. 4 and 3 1/2 e. 4 and 3 1/3 2. A distribution of 6 scores has a median of 21. If the highest score increases 3 points, the median will become __. a. 21 b. 21.5 c. 24 d. Cannot be determined without additional information. e. none of these 3. If you are told a population has a mean of 25 and a variance of 0, what must you conclude? a. Someone has made a mistake. b. There is only one element in the population. c. There are no elements in the population. d. All the elements in the population are 25. e. None of the above. 4. Which of the following measures of central tendency tends to a. be most influenced by an extreme score? b. median c. mode d. mean 5. The mean is a measure of: a. variability. b. position. c. skewness. d. central tendency. e. symmetry. 6. Suppose the manager of a plant is concerned with the total number of man-hours lost due to accidents for the past 12 months. The company statistician has reported the mean number of man-hours lost per month but did not keep a record of the total sum. Should the manager order the study repeated to obtain the desired information? Explain...
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...and Conquer? Give an example. c) (Dynamic Programming Problem: Shortest Path) Give the table to find the shortest path from node 9 to every other node. d) Part c was Dijkstra algorithm. Was it a Breath First Search Algorithm or Depth Search First Algorithm? What heap from #6 on this test is commonly used for Dijkstra’s Algorithm. e) (Dynamic Programming Problem: LCS) Suppose you have the sequence: 3, 1, 4, 7, 9, 2, 11, 13, 14. The largest increasing subsequence is 1 4 7 9 11 13 14. Use LCS to find the largest increasing subsequence. 4) (20 Points) a) Using 3c, formulate the shortest path from node 9 to node 6. Solve with Lindo. b) Using 3c, with the edges as capacity, formulate the max flow of the network. Solve with Lindo....
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...Event Analytics * Add features to website and measure its effectiveness in form of clicks, link sharing, page views * Descriptive Analytics Tools -> Google Analytics, Optimizely Diagnostic Analytics: * Post Event Analytics * Analytics used to diagnose why something/phenomenon happened the way it did * It basically provides a very good understanding of a limited piece of the problem you want to solve. * Usually less than 10% of companies surveyed do this on occasion and less than 5% do so consistently. Predictive Analytics: * Used for Prediction of Phenomenon using past and current data statistics * Essentially, you can predict what will happen if you keep things as they are. * However, less than 1% of companies surveyed have tried this yet. The ones who have, found incredible results that have already made a big difference in their business. * Eg:- SAS, RapidMiner, Statistica Prescriptive Analytics: * Prescriptive analytics automatically synthesizes big data, multiple disciplines of mathematical sciences and computational sciences, and business rules, to make predictions and then suggests decision options to take advantage of the predictions. * It is considered final phase of Analytics Some Analytics Techniques used Linear Regression In statistics, linear regression is an approach for modeling the relationship between a scalar dependent variable y and one or more explanatory variables (or independent variable) denoted...
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...hon@gmail.com> Contents 1 Descriptive Statistics 2 1.1 Descriptive vs. Inferential . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Means, Medians, and Modes . . . . . . . . . . . . . . . . . . . . . 2 1.3 Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Dispersion Percentages . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Graphs and Displays 2.1 9 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Medians, Modes, and Means Revisited . . . . . . . . . . . 10 2.1.3 z-Scores and Percentile Ranks Revisited . . . . . . . . . . 11 2.2 Stem and Leaf Displays . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Five Number Summaries and Box and Whisker Displays . . . . . 12 3 Probability 13 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.2 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.3 Variance and Standard Deviation . . . . . . . . . . . . . . 17 3.2.4 “Shortcuts” for Binomial Random Variables . . . . . . . . 18 1 4 Probability Distributions...
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...Gunter is suing ACME for $800 for retrieving information from the wrong hard drive supplied by ACME, $5,000 for time spent reconstructing the lost information, and $10,000 for punitive damages. The point of this report is to analyze the legal ramifications and statistics based on the situation at hand and to deduce the appropriate course of action to take based off the analysis. We analyzed a survey using statistics that the maximum amount ACME would have to pay in court to Gunter is $2,242.00. Based off this and our analysis of common tort law, we recommend you settle for the aforementioned amount outside of court. Table of Contents Executive Summary……………………………………………………………………….1 Table of Contents………………………………………………………………………….2 Legal...
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...Rosario Chapter 5 1. Let X be a random variable with probability density function c(1 − x2 ) −1 < x < 1 0 otherwise ∞ f (x) = (a) What is the value of c? We know that for f (x) to be a probability distribution −∞ f (x)dx = 1. We integrate f (x) with respect to x, set the result equal to 1 and solve for c. 1 1 = −1 c(1 − x2 )dx cx − c x3 3 1 −1 = = = = c = Thus, c = 3 4 c c − −c + c− 3 3 2c −2c − 3 3 4c 3 3 4 . (b) What is the cumulative distribution function of X? We want to find F (x). To do that, integrate f (x) from the lower bound of the domain on which f (x) = 0 to x so we will get an expression in terms of x. x F (x) = −1 c(1 − x2 )dx cx − cx3 3 x −1 = But recall that c = 3 . 4 3 1 3 1 = x− x + 4 4 2 = 3 4 x− x3 3 + 2 3 −1 < x < 1 elsewhere 0 1 4. The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by, 10 x2 f (x) = (a) Find P (X > 20). 0 x > 10 x ≤ 10 There are two ways to solve this problem, and other problems like it. We note that the area we are interested in is bounded below by 20 and unbounded above. Thus, ∞ P (X > c) = c f (x)dx Unlike in the discrete case, there is not really an advantage to using the complement, but you can of course do so. We could consider P (X > c) = 1 − P (X < c), c P (X > c) = 1 − P (X < c) = 1 − −∞ f (x)dx P (X > 20) = 10 dx x2 20 10 ∞ = − x 20 10 = lim − x→∞ x 1 2 ∞ −...
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...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 four computers. Complete the table...
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...advanced placement for a one-semester introductory college statistics course. Textbook: The Practice of Statistics, 3rd ed. (2008) by Yates, Moore and Starnes (Freeman Publishers) Calculator needed: TI-83 Graphing Calculator (Rentals Available) TI-83+, TI-84, TI-84+ are acceptable calculators as well Note: Any other calculator may/may not have statistical capabilities, and the instructor shall assist whenever possible, but in these instances, the student shall have sole responsibility for the calculator’s use and application in this course. AP STATISTICS Textbook: The Practice of Statistics, 3rd edition by Yates, Moore and Starnes Preliminary Chapter – What Is Statistics? (2 Days) A. Where Do Data Come From? 1. Explain why we should not draw conclusions based on personal experiences. 2. Recognize whether a study is an experiment, a survey, or an observational study that is not a survey. 3. Determine the best method for producing data to answer a specific question: experiment, survey, or other observational study. 4. Locate available data on the Internet to help you answer a question of interest. B. Dealing with Data...
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...482 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 42, NO. 2, APRIL 2012 An Adaptive Differential Evolution Algorithm With Novel Mutation and Crossover Strategies for Global Numerical Optimization Sk. Minhazul Islam, Swagatam Das, Member, IEEE, Saurav Ghosh, Subhrajit Roy, and Ponnuthurai Nagaratnam Suganthan, Senior Member, IEEE Abstract—Differential evolution (DE) is one of the most powerful stochastic real parameter optimizers of current interest. In this paper, we propose a new mutation strategy, a fitnessinduced parent selection scheme for the binomial crossover of DE, and a simple but effective scheme of adapting two of its most important control parameters with an objective of achieving improved performance. The new mutation operator, which we call DE/current-to-gr_best/1, is a variant of the classical DE/current-to-best/1 scheme. It uses the best of a group (whose size is q% of the population size) of randomly selected solutions from current generation to perturb the parent (target) vector, unlike DE/current-to-best/1 that always picks the best vector of the entire population to perturb the target vector. In our modified framework of recombination, a biased parent selection scheme has been incorporated by letting each mutant undergo the usual binomial crossover with one of the p top-ranked individuals from the current population and not with the target vector with the same index as used in all variants of DE. A DE variant obtained...
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