...Probability review (week 2) 1 Bernoulli, Binomial, Poisson and normal distributions. In this excercise we deal with Bernoulli, binomial, Poisson and normal random variables (RVs). A Bernoulli RV X models experiments, such as a coin toss, where success happens with probability p and failure with probability 1 − p. Success is indicated by X = 1 and failure by X = 0. Therefore, the probability mass function (pmf) of X is P {X = 0} = 1 − p, P {X = 1} = p (1) A binomial random variable (RV) with parameters (n, p) counts the number of successes in n independent Bernoulli trials that succeed with probability p. Thus, we can write a Binomial RV Y as n Y = i=1 Xi (2) where the Xi are Bernoulli RVs with pmfs as in (1). The pmf of a binomial RV is easily derived by noting that we have X = x for some integer x between 0 and 1 if and only there are x successful Bernoulli trials – something that happens with probability px – and n − x failed experiments – which happens with probability (1 − p)n−x – and that there are n different ways in which this could happen. x Thus n x n! px (1 − p)n−x , x = 0, 1, . . . , n. (3) p(x) := P {X = x} = p (1 − p)n−x = (n − x)!x! x A Poisson RV X takes values in the nonnegative integers. We say that X is Poisson with parameter λ it its pmf is λx p(x) = e−λ , x = 0, 1, . . . (4) x! Different from the other two, a normal random variable X can take any real value (not just 0 or 1 like the Bernoulli or integers between 0 and n for the binomial...
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...Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #07 Random Variables So, far we were discussing the laws of probability so, in the laws of the probability we have a random experiment, as a consequence of that we have a sample space, we consider a subset of the, we consider a class of subsets of the sample space which we call our event space or the events and then we define a probability function on that. Now, we consider various types of problems for example, calculating the probability of occurrence of a certain number in throwing of a die, probability of occurrence of certain card in a drain probability of various kinds of events. However, in most of the practical situations we may not be interested in the full physical description of the sample space or the events; rather we may be interested in certain numerical characteristic of the event, consider suppose I have ten instruments and they are operating for a certain amount of time, now after amount after working for a certain amount of time, we may like to know that, how many of them are actually working in a proper way and how many of them are not working properly. Now, if there are ten instruments, it may happen that seven of them are working properly and three of them are not working properly, at this stage we may not be interested in knowing the positions, suppose we are saying one instrument, two instruments and so, on tenth...
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...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, people usually do not answer frankly to such a subtle question. So, she devises the following procedure...
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...Modern Wireless Signals Earl McCune RF Communications Consulting, 2383 Pruneridge Ave., Santa Clara, CA, 95050, USA Abstract — With the evolution of wireless systems and services, the on-air signals themselves are also undergoing very significant transformations. This paper provides a survey of the active and coming-soon signal types adopted for wireless systems around the world. Focus is on modulation schemes, along with various measures used to characterize the signals before and after power amplification. Cost-benefit tradeoff information is introduced to provide perspective on this signal evolution. I. INTRODUCTION As two-way wireless communication becomes ubiquitous from relative obscurity 20 years ago, the signals used have evolved from those which are very simple to now include very complicated and high order modulations. And with economics demanding that older systems are not taken down before newer ones are installed, many of these signals must exist and operate side by side. This demands that the actual radio hardware used in any network infrastructure, as well as that in the mobile, remote, or subscriber devices, must usually be much more general purpose than optimized specifically for one signal type. In the design and test of this radio hardware it is very important to understand the fundamental characteristics of the signal(s) that it must support. With such a wide variety of signals, even the metrics used in their characterization are not uniform in type and...
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...steps necessary to run a simulation of weeks between one equipment failure and another, the downtime in days due to repair and the lost revenue over a year period of time due to the simulated equipment failure. METHODOLOGY The first step in this process is simulating the time between breakdowns. Without explicit information on frequency of breakdowns, some assumptions have to be made. In this case, the assumption is between 0 and 6 weeks with the probability of a breakdown increasing as time between breakdowns increases. I was given a probability distribution graph that showed time in weeks on the x axis from 0 to 6 with a y value of 0.33 where x = 6. The graph looked something like this: From that information, I calculated that area under the line to ensure that this could be considered a continuous distribution function. For this to be true, the area under the curve must equal 1. The calculation looks like this ½bh = ½*1/3*6 =1/6*6/1 or 1. Since we determined this is a continuous probability distribution function (CDF), we can apply the calculation f(x) = x/18, 0<= x <=6 weeks to determine the weeks between equipment failures. The translation of that function for use in Excel was x=6*SQRT(r), where r is a random number between 0 and 1. I generated pseudorandom numbers for use...
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...Engineering, Alexandria, 21544, Egypt a r t i c l e i n f o a b s t r a c t The safety performance of the nuclear power plant is a very important factor enhancing the nuclear energy option. It is vague to evaluate the nuclear power plant performance but it can be measured through measuring the safety performance of the plant. In this work, the safety of nuclear power plants is assessed by developing a “Global Safety Index” (GSI). The GSI is developed by introducing three indicators: probability of accident occurrence, performance of safety system in case of an accident occurrence (during an accident), and the consequences of the accident. The GSI is developed by tracking the performance of the safety system during a design basis accident such as loss of coolant accident (LOCA). This is done by using the PCTran simulation code in simulation a PWR LOCA and introducing four indicators: the sensation time, the response time, and the recovery time together with Core Damage Frequency (CDF). Then Fuzzy Inference System is used for obtaining the GSI. The GSI is also evaluated for the advanced types for nuclear power plants, such as AP1000, and a comparison is made between the GSI evaluated for both conventional and advanced types. © 2010 Elsevier B.V. All rights reserved. Article history: Received 28 September 2009 Received in revised form 29 April 2010 Accepted 2 July 2010 1. Introduction The world demand for energy...
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...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 because game one will not affect the result of game two. If we are trying to find the probability of picking and ace out of a deck of 52 cards, we will use hypergeometric distribution. In general, each time you pick a card it is not placed...
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...cellulosic materials into i) a bio-oil similar to crude oil ii) a synthesis gas similar to natural gas, and iii) a bio-charcoal substance. The pyrolyzer machine is currently being manufactured and tested with various types of feedstocks including corn stover and energy sorghum. The economic analysis focused on creating an automated process that integrates a transportation logistics cost optimization model with geographic information system (GIS) data. The geographic data provides possible paths for the mobile bioenergy pyrolysis unit as it moves to and from each harvest area, depending on stochastic availability of feedstock (determined by historical crop yields) and distance to oil refineries. The results indicated that there is a low probability of a positive Net Present Value (NPV) with current economic conditions. In general, the NPV was highest with a stationary scenario and it decreased with...
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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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... ------ (1) 1-a Solution Since b. let 1-b Solution ------------------------------------------------- 2. notation Page 147 in “Fundamentals of Queuing Theory –Third Edition- , Donald Gross Carl M. Harris a. b. ------------------------------------------------- ------------------------------------------------- ------------------------------------------------- 3. a. let X=service time (Random variable) and XT=total service time (Random variable) X2=X+X, X3=X+X+X, ….. f2(x2) : pdf of X2, f3(x3) : pdf of X3, …… B2(x2) : Cdf of X2, B3(x3) : Cdf of X3, …… Let B1(x)=B(x) b. Solution of 3-b c. d. Notation Proof b. Or Let F,G, and H...
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...• BLISS, FORSYTHE, AND CHAN MIMO Wireless Communication MIMO Wireless Communication Daniel W. Bliss, Keith W. Forsythe, and Amanda M. Chan ■ Wireless communication using multiple-input multiple-output (MIMO) systems enables increased spectral efficiency for a given total transmit power. Increased capacity is achieved by introducing additional spatial channels that are exploited by using space-time coding. In this article, we survey the environmental factors that affect MIMO capacity. These factors include channel complexity, external interference, and channel estimation error. We discuss examples of space-time codes, including space-time low-density parity-check codes and spacetime turbo codes, and we investigate receiver approaches, including multichannel multiuser detection (MCMUD). The ‘multichannel’ term indicates that the receiver incorporates multiple antennas by using space-time-frequency adaptive processing. The article reports the experimental performance of these codes and receivers. M - multiple-output (MIMO) systems are a natural extension of developments in antenna array communication. While the advantages of multiple receive antennas, such as gain and spatial diversity, have been known and exploited for some time [1, 2, 3], the use of transmit diversity has only been investigated recently [4, 5]. The advantages of MIMO communication, which exploits the physical channel between many transmit and receive antennas, are currently receiving significant...
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...A Five-Year Study of File-System Metadata Nitin Agrawal University of Wisconsin, Madison nitina@cs.wisc.edu William J. Bolosky, John R. Douceur, Jacob R. Lorch Microsoft Research {bolosky,johndo,lorch}@microsoft.com Abstract For five years, we collected annual snapshots of filesystem metadata from over 60,000 Windows PC file systems in a large corporation. In this paper, we use these snapshots to study temporal changes in file size, file age, file-type frequency, directory size, namespace structure, file-system population, storage capacity and consumption, and degree of file modification. We present a generative model that explains the namespace structure and the distribution of directory sizes. We find significant temporal trends relating to the popularity of certain file types, the origin of file content, the way the namespace is used, and the degree of variation among file systems, as well as more pedestrian changes in sizes and capacities. We give examples of consequent lessons for designers of file systems and related software. characteristics of file systems, including file and directory population, storage capacity, storage consumption, and degree of file modification. The contributions of this work are threefold. First, we contribute the collected data set, which we will sanitize and make available for general use later this year. This is the largest set of file-system metadata ever collected, and it spans the longest time period of any sizeable metadata collection. To obtain this...
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...loans portfolios ∗ Javier Menc´ ıa Bank of Spain June 2008 Preliminary and Incomplete Abstract This paper develops a dynamic model to assess the risk and profitability of loans portfolios. I obtain their risk premia and derive the risk-neutral measure for an exponentially affine stochastic discount factor. I employ mean-variance analysis with a VaR constraint to assess efficiency. Then I compare Spanish institutions in an empirical application, where small institutions seem to be less efficient than large ones on aggregate terms, while commercial and savings banks perform better on their respective traditional markets. Finally, I find increasing discrepancies between riskneutral and actual default probabilities since June 2007 and discuss their possible sources. Keywords: Credit risk, Probability of default, Asset Pricing, Mean-Variance allocation, Stochastic Discount Factor, Value at Risk. JEL: G21, G12, G11, C32, D81, G28. This paper is the sole responsibility of its author. The views represented here do not necessarily reflect those of the Bank of Spain. Thanks are due to Alfredo Mart´ for his valuable suggestions as well as for ın, help with the interest rate database. Of course, the usual caveat applies. Address for correspondence: Alcal´ 48, E-28014 Madrid, Spain, tel: +34 91 338 5414, fax: +34 91 338 6102. a ∗ 1 Introduction Standard capital market theory states that there is a risk-return tradeoff in equilib- rium. The more risk one is willing to take, the higher...
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...1. BASIC STATS Types of variables: Count: no of bedrooms/children, manufacture year Ordinal: categories, income brackets Nominal: gender, yes/no, manufacture model Continuous: distance, time, age, best cruising speed Graphs Scatterplots: shows if there is a linear relationship (positive increasing left to right, cov>0), doesn't measure strength Histogram: modality skewness (positive long tail to right), modal class, symmetry Box Plot: Skewness (short/long whisker, short bottom=positive), symmetric Empirical CDFs | |Sampl|Populat| | |e |ion | |Average/Mean |[pic]|μ | |Variance | s2 |σ2 | |Standard |s |Σ | |Deviation | | | |Correlation |r |ρ | Location of Percentiles = ___th observation = what figure? Arithmetic Mean Mean vs Median If symmetric, mean ≈ median If positive skew, mean > median If negative skew, mean < median Variance Measures spread Larger SD – ↑ risk – ↑ rate of return Eg Standard Deviation Coefficient of variation Measures spread Covariance Measures the strength (and direction) of linear relationship between 2 variables. If cov > 0, then as X increases, Y increases (positive slope). If cov < 0 = opposite. If cov=0, not linearly related Coefficient of correlation If r=-1, perfect negative linear relationship If r=+1, perfect positive relationship If r=0, no LINEAR relationship From...
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...FACTORS INFLUENCING IMPLEMENTION OF CONSTITUENCY DEVELOPMENT FUND PROJECTS IN GARISA COUNTY BY ABDIKADIR ADEN FARAH A RESEARCH PROPOSAL SUBMITTED TO THE SCHOOL OF SOCIAL SCIENCES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF MASTER OF GOVERNANCE AND ETHICS OF MOUNT KENYA UNIVERSITY OCTOBER, 2014 DECLARATION This research proposal is my original work and has not been presented for a degree in any other university or for any other award. No part of this study should be reproduced without authority of the author or/and of Mount Kenya University. Signature:_______________ Date:________________ Abdikadir Aden Farah MGE (DL) 111/23452 This research proposal has been presented for examination with our approval as the university supervisors Signature:______________ Date:________________ Mr. Godfrey Kinyua School of Social Sciences Mount Kenya University Signature:_______________ Date: _________________ Prof. Geoffrey Owino School of Social Sciences Mount Kenya University DEDICATION This study is dedicated to my dear family for their love, understanding and support during the many long hours committed to this program. ACKNOWLEDGEMENT I am highly indebted to my supervisors, Mr. Godfrey Kinyua for his availability, and also for providing the necessary guidance through each and every stage of this proposal. I am grateful to the Library staff of Mount Kenya University, for helping...
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